Question 1: Palette 1/145

JEE Mains 2026 MCQ

Let the circle $x^2 + y^2 = 4$ intersect x-axis at the points A$(a, 0)$, $a > 0$ and B$(b, 0)$. Let $P(2 \cos \alpha, 2 \sin \alpha)$, $0 < \alpha < \frac{\pi}{2}$ and $Q(2 \cos \beta, 2 \sin \beta)$ be two points such that $(\alpha - \beta) = \frac{\pi}{2}$. Then the point of intersection of AQ and BP lies on :

A

$x^2 + y^2 - 4x - 4 = 0$

B

$x^2 + y^2 - 4x - 4y = 0$

C

$x^2 + y^2 - 4x - 4y - 4 = 0$

D

$x^2 + y^2 - 4y - 4 = 0$

Question 2: Palette 2/145

JEE Mains 2026 MCQ

Let $y=x$ be the equation of a chord of the circle $\mathrm{C}_1$ (in the closed half-plane $x \geq 0$ ) of diameter 10 passing through the origin. Let $\mathrm{C}_2$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $\mathrm{C}_2$, which passes through the point $(2,3)$ and is farthest from the center of $\mathrm{C}_2$, is $x+a y+b=0$, then $a-b$ is equal to

A

-6

B

10

C

6

D

-2

Question 3: Palette 3/145

JEE Mains 2026 MCQ

Let a circle of radius 4 pass through the origin O , the points $\mathrm{A}(-\sqrt{3} a, 0)$ and $\mathrm{B}(0,-\sqrt{2} b)$, where $a$ and $b$ are real parameters and $a b \neq 0$. Then the locus of the centroid of $\triangle \mathrm{OAB}$ is a circle of radius

A

$\frac{7}{3}$

B

$\frac{11}{3}$

C

$\frac{5}{3}$

D

$\frac{8}{3}$

Question 4: Palette 4/145

JEE Mains 2026 MCQ

Let the set of all values of $r$, for which the circles $(x+1)^2+(y+4)^2=r^2$ and $x^2+y^2-4 x-2 y-4=0$ intersect at two distinct points be the interval $(\alpha, \beta)$. Then $\alpha \beta$ is equal to

A

21

B

24

C

20

D

25

Question 5: Palette 5/145

JEE Mains 2026 MCQ

Let PQ and MN be two straight lines touching the circle $x^2+y^2-4 x-6 y-3=0$ at the points $A$ and $B$ respectively. Let $O$ be the centre of the circle and $\angle A O B=\pi / 3$. Then the locus of the point of intersection of the lines PQ and MN is :

A

$x^2+y^2-18 x-12 y-25=0$

B

$x^2+y^2-12 x-18 y-25=0$

C

$3\left(x^2+y^2\right)-12 x-18 y-25=0$

D

$3\left(x^2+y^2\right)-18 x-12 y+25=0$

Question 6: Palette 6/145

JEE Mains 2026 MCQ

Let C be a circle having centre in the first quadrant and touching the $x$-axis at a distance of 3 units from the origin. If the circle $C$ has an intercept of length $6 \sqrt{3}$ on $y$-axis, then the length of the chord of the circle C on the line $x-y=3$ is :

A

${ }8$

B

${ }6$

C

$6 \sqrt{2}$

D

$ 8 \sqrt{2} $

Question 7: Palette 7/145

JEE Mains 2026 MCQ

Let the point P be the vertex of the parabola $y=x^2-6 x+12$. If a line passing through the point P intersects the circle $x^2+y^2-2 x-4 y+3=0$ at the points R and S , then the maximum value of $(\mathrm{PR}+\mathrm{PS})^2$ is :

A

10

B

20

C

25

D

5

Question 8: Palette 8/145

JEE Mains 2026 MCQ

Let P be a moving point on the circle $x^2+y^2-6 x-8 y+21=0$. Then, the maximum distance of P from the vertex of the parabola $x^2+6 x+y+13=0$ is equal to:

A

8

B

10

C

12

D

9

Question 9: Palette 9/145

JEE Mains 2026 MCQ

Suppose that two chords, drawn from the point $(1,2)$ on the circle $x^2+y^2+x-3 y=0$ are bisected by the $y$-axis. If the other ends of these chords are R and S , and the mid point of the line segment RS is $(\alpha, \beta)$, then $6(\alpha+\beta)$ is equal to :

A

1

B

3

C

4

D

6

Question 10: Palette 10/145

JEE Mains 2026 MCQ

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines $x + (k-1)y + 3 = 0$ and $2x + k^2y - 4 = 0$. If the line $x - y + 2 = 0$ intersects the circle at the points A and B, then $(AB)^2$ is equal to :

A

10

B

27

C

18

D

34

Question 11: Palette 11/145

JEE Mains 2025 MCQ

Let $C_1$ be the circle in the third quadrant of radius 3 , that touches both coordinate axes. Let $C_2$ be the circle with centre $(1,3)$ that touches $\mathrm{C}_1$ externally at the point $(\alpha, \beta)$. If $(\beta-\alpha)^2=\frac{m}{n}$ , $\operatorname{gcd}(m, n)=1$, then $m+n$ is equal to

A

22

B

13

C

9

D

31

Question 12: Palette 12/145

JEE Mains 2025 MCQ

If the four distinct points $(4,6),(-1,5),(0,0)$ and $(k, 3 k)$ lie on a circle of radius $r$, then $10 k+r^2$ is equal to

A

34

B

32

C

35

D

33

Question 13: Palette 13/145

JEE Mains 2025 MCQ

Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on 3x + 2y + 2 = 0. Then the length of the chord, of the circle C, whose mid-point is (1, 2), is:

A

4$\sqrt{2}$

B

2$\sqrt{2}$

C

2$\sqrt{3}$

D

$\sqrt{3}$

Question 14: Palette 14/145

JEE Mains 2025 MCQ

Let the line x+y=1 meet the circle $x^2+y^2=4$ at the points A and B. If the line perpendicular to AB and passing through the mid-point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ABCD is equal to :

A

$ \sqrt{14} $

B

$ 3\sqrt{7} $

C

$ 2\sqrt{14} $

D

$ 5\sqrt{7} $

Question 15: Palette 15/145

JEE Mains 2025 MCQ

Let the equation of the circle, which touches $x$-axis at the point $(a, 0), a>0$ and cuts off an intercept of length $b$ on $y-a x i s$ be $x^2+y^2-\alpha x+\beta y+\gamma=0$. If the circle lies below $x-a x i s$, then the ordered pair $\left(2 a, b^2\right)$ is equal to

A

$\left(\alpha, \beta^2+4 \gamma\right)$

B

$\left(\alpha, \beta^2-4 \gamma\right)$

C

$\left(\gamma, \beta^2-4 \alpha\right)$

D

$\left(\gamma, \beta^2+4 \alpha\right)$

Question 16: Palette 16/145

JEE Mains 2025 MCQ

Let circle $C$ be the image of $x^2+y^2-2 x+4 y-4=0$ in the line $2 x-3 y+5=0$ and $A$ be the point on $C$ such that $O A$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta<4$, lies on $C$ such that the length of the arc $A B$ is $(1 / 6)^{\text {th }}$ of the perimeter of $C$, then $\beta-\sqrt{3} \alpha$ is equal to

A

$4-\sqrt{3}$

B

 $3$

C

$4$

D

$3+\sqrt{3}$

Question 17: Palette 17/145

JEE Mains 2025 MCQ

A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point $(2,5)$ and intersects the circle $C$ at exactly two points. If the set of all possible values of r is the interval $(\alpha, \beta)$, then $3 \beta-2 \alpha$ is equal to :

A

10

B

12

C

14

D

15

Question 18: Palette 18/145

JEE Mains 2024 MCQ

Let a circle passing through $(2,0)$ have its centre at the point $(\mathrm{h}, \mathrm{k})$. Let $(x_{\mathrm{c}}, y_{\mathrm{c}})$ be the point of intersection of the lines $3 x+5 y=1$ and $(2+\mathrm{c}) x+5 \mathrm{c}^2 y=1$. If $\mathrm{h}=\lim _\limits{\mathrm{c} \rightarrow 1} x_{\mathrm{c}}$ and $\mathrm{k}=\lim _\limits{\mathrm{c} \rightarrow 1} y_{\mathrm{c}}$, then the equation of the circle is :

A

$5 x^2+5 y^2-4 x-2 y-12=0$

B

$25 x^2+25 y^2-20 x+2 y-60=0$

C

$25 x^2+25 y^2-2 x+2 y-60=0$

D

$5 x^2+5 y^2-4 x+2 y-12=0$

Question 19: Palette 19/145

JEE Mains 2024 MCQ

If the image of the point $(-4,5)$ in the line $x+2 y=2$ lies on the circle $(x+4)^2+(y-3)^2=r^2$, then $r$ is equal to:

A

2

B

3

C

4

D

1

Question 20: Palette 20/145

JEE Mains 2024 MCQ

Let the circles $C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2$ and $C_2:(x-8)^2+\left(y-\frac{15}{2}\right)^2=r_2^2$ touch each other externally at the point $(6,6)$. If the point $(6,6)$ divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio $2: 1$, then $(\alpha+\beta)+4\left(r_1^2+r_2^2\right)$ equals

A

130

B

110

C

145

D

125

Question 21: Palette 21/145

JEE Mains 2024 MCQ

If $\mathrm{P}(6,1)$ be the orthocentre of the triangle whose vertices are $\mathrm{A}(5,-2), \mathrm{B}(8,3)$ and $\mathrm{C}(\mathrm{h}, \mathrm{k})$, then the point $\mathrm{C}$ lies on the circle :

A

$x^2+y^2-74=0$

B

$x^2+y^2-65=0$

C

$x^2+y^2-61=0$

D

$x^2+y^2-52=0$

Question 22: Palette 22/145

JEE Mains 2024 MCQ

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m+n^2$ is equal to

A

408

B

414

C

312

D

396

Question 23: Palette 23/145

JEE Mains 2024 MCQ

Let the circle $C_1: x^2+y^2-2(x+y)+1=0$ and $\mathrm{C_2}$ be a circle having centre at $(-1,0)$ and radius 2 . If the line of the common chord of $\mathrm{C}_1$ and $\mathrm{C}_2$ intersects the $\mathrm{y}$-axis at the point $\mathrm{P}$, then the square of the distance of P from the centre of $\mathrm{C_1}$ is:

A

4

B

6

C

2

D

1

Question 24: Palette 24/145

JEE Mains 2024 MCQ

Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies :

A

$\mathrm{r}=1$

B

$2 \mathrm{r}^2-4 \mathrm{r}+1=0$

C

$2 \mathrm{r}^2-8 \mathrm{r}+7=0$

D

$\mathrm{r}^2-8 \mathrm{r}+8=0$

Question 25: Palette 25/145

JEE Mains 2024 MCQ

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point $(3,2)$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point $(5,5)$ is :

A

4$\sqrt2$

B

4

C

5

D

2$\sqrt2$

Question 26: Palette 26/145

JEE Mains 2024 MCQ

Let $\mathrm{C}$ be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x+y=2$ intersects the circle $\mathrm{C}$ at the points $\mathrm{P}$ and $\mathrm{Q}$. Let $\mathrm{MN}$ be a chord of $\mathrm{C}$ of length 2 unit and slope $-1$. Then, a distance (in units) between the chord PQ and the chord $\mathrm{MN}$ is

A

$3-\sqrt{2}$

B

$2-\sqrt{3}$

C

$\sqrt{2}-1$

D

$\sqrt{2}+1$

Question 27: Palette 27/145

JEE Mains 2024 MCQ

A square is inscribed in the circle $x^2+y^2-10 x-6 y+30=0$. One side of this square is parallel to $y=x+3$. If $\left(x_i, y_i\right)$ are the vertices of the square, then $\Sigma\left(x_i^2+y_i^2\right)$ is equal to:

A

152

B

148

C

156

D

160

Question 28: Palette 28/145

JEE Mains 2024 MCQ

Let the locus of the midpoints of the chords of the circle $x^2+(y-1)^2=1$ drawn from the origin intersect the line $x+y=1$ at $\mathrm{P}$ and $\mathrm{Q}$. Then, the length of $\mathrm{PQ}$ is :

A

$\frac{1}{2}$

B

1

C

$\frac{1}{\sqrt{2}}$

D

$\sqrt{2}$

Question 29: Palette 29/145

JEE Mains 2024 MCQ

Let $C: x^2+y^2=4$ and $C^{\prime}: x^2+y^2-4 \lambda x+9=0$ be two circles. If the set of all values of $\lambda$ so that the circles $\mathrm{C}$ and $\mathrm{C}$ intersect at two distinct points, is $\mathrm{R}-[\mathrm{a}, \mathrm{b}]$, then the point $(8 \mathrm{a}+12,16 \mathrm{~b}-20)$ lies on the curve :

A

$x^2+2 y^2-5 x+6 y=3$

B

$5 x^2-y=-11$

C

$x^2-4 y^2=7$

D

$6 x^2+y^2=42$

Question 30: Palette 30/145

JEE Mains 2024 MCQ

Let a variable line passing through the centre of the circle $x^2+y^2-16 x-4 y=0$, meet the positive co-ordinate axes at the points $A$ and $B$. Then the minimum value of $O A+O B$, where $O$ is the origin, is equal to

A

12

B

20

C

24

D

18

Question 31: Palette 31/145

JEE Mains 2024 MCQ

If one of the diameters of the circle $x^2+y^2-10 x+4 y+13=0$ is a chord of another circle $\mathrm{C}$, whose center is the point of intersection of the lines $2 x+3 y=12$ and $3 x-2 y=5$, then the radius of the circle $\mathrm{C}$ is :

A

4

B

3$\sqrt2$

C

6

D

$\sqrt{20}$

Question 32: Palette 32/145

JEE Mains 2024 MCQ

If the circles $(x+1)^2+(y+2)^2=r^2$ and $x^2+y^2-4 x-4 y+4=0$ intersect at exactly two distinct points, then

A

$\frac{1}{2}<\mathrm{r}<7$

B

$3<\mathrm{r}<7$

C

$5<\mathrm{r}<9$

D

$0<\mathrm{r}<7$

Question 33: Palette 33/145

JEE Mains 2024 MCQ

Four distinct points $(2 k, 3 k),(1,0),(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to :

A

$\frac{3}{13}$

B

$\frac{2}{13}$

C

$\frac{5}{13}$

D

$\frac{1}{13}$

Question 34: Palette 34/145

JEE Mains 2023 MCQ

The number of common tangents, to the circles

$x^{2}+y^{2}-18 x-15 y+131=0$

and $x^{2}+y^{2}-6 x-6 y-7=0$, is :

A

4

B

2

C

3

D

1

Question 35: Palette 35/145

JEE Mains 2023 MCQ

Let the centre of a circle C be $(\alpha, \beta)$ and its radius $r < 8$. Let $3 x+4 y=24$ and $3 x-4 y=32$ be two tangents and $4 x+3 y=1$ be a normal to C. Then $(\alpha-\beta+r)$ is equal to :

A

7

B

9

C

5

D

6

Question 36: Palette 36/145

JEE Mains 2023 MCQ

Let A be the point $(1,2)$ and B be any point on the curve $x^{2}+y^{2}=16$. If the centre of the locus of the point P, which divides the line segment $\mathrm{AB}$ in the ratio $3: 2$ is the point C$(\alpha, \beta)$, then the length of the line segment $\mathrm{AC}$ is :

A

$\frac{3 \sqrt{5}}{5}$

B

$\frac{6 \sqrt{5}}{5}$

C

$\frac{2 \sqrt{5}}{5}$

D

$\frac{4 \sqrt{5}}{5}$

Question 37: Palette 37/145

JEE Mains 2023 MCQ

A line segment AB of length $\lambda$ moves such that the points A and B remain on the periphery of a circle of radius $\lambda$. Then the locus of the point, that divides the line segment AB in the ratio 2 : 3, is a circle of radius :

A

${2 \over 3}\lambda $

B

${3 \over 5}\lambda $

C

${{\sqrt {19} } \over 7}\lambda $

D

${{\sqrt {19} } \over 5}\lambda $

Question 38: Palette 38/145

JEE Mains 2023 MCQ

Let O be the origin and OP and OQ be the tangents to the circle $x^2+y^2-6x+4y+8=0$ at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point $\left( {\alpha ,{1 \over 2}} \right)$, then a value of $\alpha$ is :

A

1

B

$-\frac{1}{2}$

C

$\frac{5}{2}$

D

$\frac{3}{2}$

Question 39: Palette 39/145

JEE Mains 2023 MCQ

If the tangents at the points $\mathrm{P}$ and $\mathrm{Q}$ on the circle $x^{2}+y^{2}-2 x+y=5$ meet at the point $R\left(\frac{9}{4}, 2\right)$, then the area of the triangle $\mathrm{PQR}$ is :

A

$\frac{13}{8}$

B

$\frac{5}{8}$

C

$\frac{5}{4}$

D

$\frac{13}{4}$

Question 40: Palette 40/145

JEE Mains 2023 MCQ

The set of all values of $a^{2}$ for which the line $x+y=0$ bisects two distinct chords drawn from a point $\mathrm{P}\left(\frac{1+a}{2}, \frac{1-a}{2}\right)$ on the circle $2 x^{2}+2 y^{2}-(1+a) x-(1-a) y=0$, is equal to :

A

$(0,4]$

B

$(4, \infty)$

C

$(2,12]$

D

$(8, \infty)$

Question 41: Palette 41/145

JEE Mains 2023 MCQ

Let a circle $C_{1}$ be obtained on rolling the circle $x^{2}+y^{2}-4 x-6 y+11=0$ upwards 4 units on the tangent $\mathrm{T}$ to it at the point $(3,2)$. Let $C_{2}$ be the image of $C_{1}$ in $\mathrm{T}$. Let $A$ and $B$ be the centers of circles $C_{1}$ and $C_{2}$ respectively, and $M$ and $N$ be respectively the feet of perpendiculars drawn from $A$ and $B$ on the $x$-axis. Then the area of the trapezium AMNB is :

A

$2\left( {2 + \sqrt 2 } \right)$

B

$4\left( {1 + \sqrt 2 } \right)$

C

$3 + 2\sqrt 2 $

D

$2\left( {1 + \sqrt 2 } \right)$

Question 42: Palette 42/145

JEE Mains 2023 MCQ

Let $y=x+2,4y=3x+6$ and $3y=4x+1$ be three tangent lines to the circle $(x-h)^2+(y-k)^2=r^2$. Then $h+k$ is equal to :

A

6

B

5 (1 + $\sqrt2$)

C

5

D

5$\sqrt2$

Question 43: Palette 43/145

JEE Mains 2023 MCQ

Let the tangents at the points $A(4,-11)$ and $B(8,-5)$ on the circle $x^{2}+y^{2}-3 x+10 y-15=0$, intersect at the point $C$. Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to :

A

$\frac{2\sqrt{13}}{3}$

B

$\frac{3\sqrt{3}}{4}$

C

$\sqrt{13}$

D

$2\sqrt{13}$

Question 44: Palette 44/145

JEE Mains 2023 MCQ

The points of intersection of the line $ax + by = 0,(a \ne b)$ and the circle ${x^2} + {y^2} - 2x = 0$ are $A(\alpha ,0)$ and $B(1,\beta )$. The image of the circle with AB as a diameter in the line $x + y + 2 = 0$ is :

A

${x^2} + {y^2} + 5x + 5y + 12 = 0$

B

${x^2} + {y^2} + 3x + 5y + 8 = 0$

C

${x^2} + {y^2} - 5x - 5y + 12 = 0$

D

${x^2} + {y^2} + 3x + 3y + 4 = 0$

Question 45: Palette 45/145

JEE Mains 2023 MCQ

The locus of the mid points of the chords of the circle ${C_1}:{(x - 4)^2} + {(y - 5)^2} = 4$ which subtend an angle ${\theta _i}$ at the centre of the circle $C_1$, is a circle of radius $r_i$. If ${\theta _1} = {\pi \over 3},{\theta _3} = {{2\pi } \over 3}$ and $r_1^2 = r_2^2 + r_3^2$, then ${\theta _2}$ is equal to :

A

${\pi \over 2}$

B

${\pi \over 4}$

C

${{3\pi } \over 4}$

D

${\pi \over 6}$

Question 46: Palette 46/145

JEE Mains 2022 MCQ

Let the tangents at two points $\mathrm{A}$ and $\mathrm{B}$ on the circle $x^{2}+\mathrm{y}^{2}-4 x+3=0$ meet at origin $\mathrm{O}(0,0)$. Then the area of the triangle $\mathrm{OAB}$ is :

A

$\frac{3 \sqrt{3}}{2}$

B

$\frac{3 \sqrt{3}}{4}$

C

$\frac{3}{2 \sqrt{3}}$

D

$\frac{3}{4 \sqrt{3}}$

Question 47: Palette 47/145

JEE Mains 2022 MCQ

For $\mathrm{t} \in(0,2 \pi)$, if $\mathrm{ABC}$ is an equilateral triangle with vertices $\mathrm{A}(\sin t,-\cos \mathrm{t}), \mathrm{B}(\operatorname{cost}, \sin t)$ and $C(a, b)$ such that its orthocentre lies on a circle with centre $\left(1, \frac{1}{3}\right)$, then $\left(a^{2}-b^{2}\right)$ is equal to :

A

$\frac{8}{3}$

B

8

C

$\frac{77}{9}$

D

$\frac{80}{9}$

Question 48: Palette 48/145

JEE Mains 2022 MCQ

Let $C$ be the centre of the circle $x^{2}+y^{2}-x+2 y=\frac{11}{4}$ and $P$ be a point on the circle. A line passes through the point $\mathrm{C}$, makes an angle of $\frac{\pi}{4}$ with the line $\mathrm{CP}$ and intersects the circle at the points $Q$ and $R$. Then the area of the triangle $P Q R$ (in unit $^{2}$ ) is :

A

2

B

2$\sqrt2$

C

$8 \sin \left(\frac{\pi}{8}\right)$

D

$8 \cos \left(\frac{\pi}{8}\right)$

Question 49: Palette 49/145

JEE Mains 2022 MCQ

A circle $C_{1}$ passes through the origin $\mathrm{O}$ and has diameter 4 on the positive $x$-axis. The line $y=2 x$ gives a chord $\mathrm{OA}$ of circle $\mathrm{C}_{1}$. Let $\mathrm{C}_{2}$ be the circle with $\mathrm{OA}$ as a diameter. If the tangent to $\mathrm{C}_{2}$ at the point $\mathrm{A}$ meets the $x$-axis at $\mathrm{P}$ and $y$-axis at $\mathrm{Q}$, then $\mathrm{QA}: \mathrm{AP}$ is equal to :

A

1 : 4

B

1 : 5

C

2 : 5

D

1 : 3

Question 50: Palette 50/145

JEE Mains 2022 MCQ

If the circle $x^{2}+y^{2}-2 g x+6 y-19 c=0, g, c \in \mathbb{R}$ passes through the point $(6,1)$ and its centre lies on the line $x-2 c y=8$, then the length of intercept made by the circle on $x$-axis is :

A

$\sqrt{11}$

B

4

C

3

D

$2 \sqrt{23}$

Question 51: Palette 51/145

JEE Mains 2022 MCQ

Let the abscissae of the two points $P$ and $Q$ on a circle be the roots of $x^{2}-4 x-6=0$ and the ordinates of $\mathrm{P}$ and $\mathrm{Q}$ be the roots of $y^{2}+2 y-7=0$. If $\mathrm{PQ}$ is a diameter of the circle $x^{2}+y^{2}+2 a x+2 b y+c=0$, then the value of $(a+b-c)$ is _____________.

A

12

B

13

C

14

D

16

Question 52: Palette 52/145

JEE Mains 2022 MCQ

Consider three circles:

${C_1}:{x^2} + {y^2} = {r^2}$

${C_2}:{(x - 1)^2} + {(y - 1)^2} = {r^2}$

${C_3}:{(x - 2)^2} + {(y - 1)^2} = {r^2}$

If a line L : y = mx + c be a common tangent to C₁, C₂ and C₃ such that C₁ and C₃ lie on one side of line L while C₂ lies on other side, then the value of $20({r^2} + c)$ is equal to :

A

23

B

15

C

12

D

6

Question 53: Palette 53/145

JEE Mains 2022 MCQ

Let a triangle ABC be inscribed in the circle ${x^2} - \sqrt 2 (x + y) + {y^2} = 0$ such that $\angle BAC = {\pi \over 2}$. If the length of side AB is $\sqrt 2 $, then the area of the $\Delta$ABC is equal to :

A

1

B

$\left( {\sqrt 6 + \sqrt 3 } \right)/2$

C

$\left( {3 + \sqrt 3 } \right)/4$

D

$\left( {\sqrt 6 + 2\sqrt 3 } \right)/4$

Question 54: Palette 54/145

JEE Mains 2022 MCQ

Let the tangent to the circle C₁ : x² + y² = 2 at the point M($-$1, 1) intersect the circle C₂ : (x $-$ 3)² + (y $-$ 2)² = 5, at two distinct points A and B. If the tangents to C₂ at the points A and B intersect at N, then the area of the triangle ANB is equal to :

A

${1 \over 2}$

B

${2 \over 3}$

C

${1 \over 6}$

D

${5 \over 3}$

Question 55: Palette 55/145

JEE Mains 2022 MCQ

If the tangents drawn at the points $O(0,0)$ and $P\left( {1 + \sqrt 5 ,2} \right)$ on the circle ${x^2} + {y^2} - 2x - 4y = 0$ intersect at the point Q, then the area of the triangle OPQ is equal to :

A

${{3 + \sqrt 5 } \over 2}$

B

${{4 + 2\sqrt 5 } \over 2}$

C

${{5 + 3\sqrt 5 } \over 2}$

D

${{7 + 3\sqrt 5 } \over 2}$

Question 56: Palette 56/145

JEE Mains 2022 MCQ

The set of values of k, for which the circle $C:4{x^2} + 4{y^2} - 12x + 8y + k = 0$ lies inside the fourth quadrant and the point $\left( {1, - {1 \over 3}} \right)$ lies on or inside the circle C, is :

A

an empty set

B

$\left( {6,{{65} \over 9}} \right]$

C

$\left[ {{{80} \over 9},10} \right)$

D

$\left( {9,{{92} \over 9}} \right]$

Question 57: Palette 57/145

JEE Mains 2022 MCQ

Let C be a circle passing through the points A(2, $-$1) and B(3, 4). The line segment AB s not a diameter of C. If r is the radius of C and its centre lies on the circle ${(x - 5)^2} + {(y - 1)^2} = {{13} \over 2}$, then r² is equal to :

A

32

B

${{65} \over 2}$

C

${{61} \over 2}$

D

30

Question 58: Palette 58/145

JEE Mains 2022 MCQ

A circle touches both the y-axis and the line x + y = 0. Then the locus of its center is :

A

$y = \sqrt 2 x$

B

$x = \sqrt 2 y$

C

${y^2} - {x^2} = 2xy$

D

${x^2} - {y^2} = 2xy$

Question 59: Palette 59/145

JEE Mains 2022 MCQ

Let a circle C touch the lines ${L_1}:4x - 3y + {K_1} = 0$ and ${L_2} = 4x - 3y + {K_2} = 0$, ${K_1},{K_2} \in R$. If a line passing through the centre of the circle C intersects L₁ at $( - 1,2)$ and L₂ at $(3, - 6)$, then the equation of the circle C is :

A

${(x - 1)^2} + {(y - 2)^2} = 4$

B

${(x + 1)^2} + {(y - 2)^2} = 4$

C

${(x - 1)^2} + {(y + 2)^2} = 16$

D

${(x - 1)^2} + {(y - 2)^2} = 16$

Question 60: Palette 60/145

JEE Mains 2021 MCQ

Let Z be the set of all integers,

$A = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {y^2} \le 4\} $

$B = \{ (x,y) \in Z \times Z:{x^2} + {y^2} \le 4\} $

$C = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {(y - 2)^2} \le 4\} $

If the total number of relation from A $\cap$ B to A $\cap$ C is 2^p, then the value of p is :

A

16

B

25

C

49

D

9

Question 61: Palette 61/145

JEE Mains 2021 MCQ

A circle C touches the line x = 2y at the point (2, 1) and intersects the circle

C₁ : x² + y² + 2y $-$ 5 = 0 at two points P and Q such that PQ is a diameter of C₁. Then the diameter of C is :

A

$7\sqrt 5 $

B

15

C

$\sqrt {285} $

D

$4\sqrt {15} $

Question 62: Palette 62/145

JEE Mains 2021 MCQ

If a line along a chord of the circle 4x² + 4y² + 120x + 675 = 0, passes through the point ($-$30, 0) and is tangent to the parabola y² = 30x, then the length of this chord is :

A

5

B

7

C

5${\sqrt 3 }$

D

3${\sqrt 5 }$

Question 63: Palette 63/145

JEE Mains 2021 MCQ

Consider a circle C which touches the y-axis at (0, 6) and cuts off an intercept $6\sqrt 5 $ on the x-axis. Then the radius of the circle C is equal to :

A

$\sqrt {53} $

B

9

C

8

D

$\sqrt {82} $

Question 64: Palette 64/145

JEE Mains 2021 MCQ

Two tangents are drawn from the point P($-$1, 1) to the circle x² + y² $-$ 2x $-$ 6y + 6 = 0. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to :

A

2

B

$(3\sqrt 2 + 2)$

C

4

D

$3(\sqrt 2 - 1)$

Question 65: Palette 65/145

JEE Mains 2021 MCQ

Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set {P, Q} is equal to :

A

{(4, 0), (0, 6)}

B

$\{ (2 + 2\sqrt 2 ,3 - \sqrt 5 ),(2 - 2\sqrt 2 ,3 + \sqrt 5 )\} $

C

$\{ (2 + 2\sqrt 2 ,3 + \sqrt 5 ),(2 - 2\sqrt 2 ,3 - \sqrt 5 )\} $

D

{($-$1, 5), (5, 1)}

Question 66: Palette 66/145

JEE Mains 2021 MCQ

Let $A = \{ (x,y) \in R \times R|2{x^2} + 2{y^2} - 2x - 2y = 1\} $, $B = \{ (x,y) \in R \times R|4{x^2} + 4{y^2} - 16y + 7 = 0\} $ and $C = \{ (x,y) \in R \times R|{x^2} + {y^2} - 4x - 2y + 5 \le {r^2}\} $.

Then the minimum value of |r| such that $A \cup B \subseteq C$ is equal to

A

${{3 + \sqrt {10} } \over 2}$

B

${{2 + \sqrt {10} } \over 2}$

C

${{3 + 2\sqrt 5 } \over 2}$

D

$1 + \sqrt 5 $

Question 67: Palette 67/145

JEE Mains 2021 MCQ

Let the circle S : 36x² + 36y² $-$ 108x + 120y + C = 0 be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, x $-$ 2y = 4 and 2x $-$ y = 5 lies inside the circle S, then :

A

${{25} \over 9} < C < {{13} \over 3}$

B

100 < C < 165

C

81 < C < 156

D

100 < C < 156

Question 68: Palette 68/145

JEE Mains 2021 MCQ

Let r₁ and r₂ be the radii of the largest and smallest circles, respectively, which pass through the point ($-$4, 1) and having their centres on the circumference of the circle x² + y² + 2x + 4y $-$ 4 = 0. If ${{{r_1}} \over {{r_2}}} = a + b\sqrt 2 $, then a + b is equal to :

A

3

B

11

C

5

D

7

Question 69: Palette 69/145

JEE Mains 2021 MCQ

Let S₁ : x² + y² = 9 and S₂ : (x $-$ 2)² + y² = 1. Then the locus of center of a variable circle S which touches S₁ internally and S₂ externally always passes through the points :

A

$\left( {{1 \over 2}, \pm {{\sqrt 5 } \over 2}} \right)$

B

(1, $\pm$ 2)

C

$\left( {2, \pm {3 \over 2}} \right)$

D

(0, $\pm$ $\sqrt 3 $)

Question 70: Palette 70/145

JEE Mains 2021 MCQ

Choose the correct statement about two circles whose equations are given below :

x² + y² $-$ 10x $-$ 10y + 41 = 0

x² + y² $-$ 22x $-$ 10y + 137 = 0

A

circles have same centre

B

circles have no meeting point

C

circles have only one meeting point

D

circles have two meeting points

Question 71: Palette 71/145

JEE Mains 2021 MCQ

For the four circles M, N, O and P, following four equations are given :

Circle M : x² + y² = 1

Circle N : x² + y² $-$ 2x = 0

Circle O : x² + y² $-$ 2x $-$ 2y + 1 = 0

Circle P : x² + y² $-$ 2y = 0

If the centre of circle M is joined with centre of the circle N, further center of circle N is joined with centre of the circle O, centre of circle O is joined with the centre of circle P and lastly, centre of circle P is joined with centre of circle M, then these lines form the sides of a :

A

Rhombus

B

Square

C

Rectangle

D

Parallelogram

Question 72: Palette 72/145

JEE Mains 2021 MCQ

Let the tangent to the circle x² + y² = 25 at the point R(3, 4) meet x-axis and y-axis at points P and Q, respectively. If r is the radius of the circle passing through the origin O and having centre at the incentre of the triangle OPQ, then r² is equal to :

A

${{585} \over {66}}$

B

${{625} \over {72}}$

C

${{529} \over {64}}$

D

${{125} \over {72}}$

Question 73: Palette 73/145

JEE Mains 2021 MCQ

Two tangents are drawn from a point P to the circle x² + y² $-$ 2x $-$ 4y + 4 = 0, such that the angle between these tangents is ${\tan ^{ - 1}}\left( {{{12} \over 5}} \right)$, where ${\tan ^{ - 1}}\left( {{{12} \over 5}} \right)$ $\in$(0, $\pi$). If the centre of the circle is denoted by C and these tangents touch the circle at points A and B, then the ratio of the areas of $\Delta$PAB and $\Delta$CAB is :

A

3 : 1

B

9 : 4

C

2 : 1

D

11 : 4

Question 74: Palette 74/145

JEE Mains 2021 MCQ

The line 2x $-$ y + 1 = 0 is a tangent to the circle at the point (2, 5) and the centre of the circle lies on x $-$ 2y = 4. Then, the radius of the circle is :

A

5$\sqrt 3 $

B

4$\sqrt 5 $

C

3$\sqrt 5 $

D

5$\sqrt 4 $

Question 75: Palette 75/145

JEE Mains 2021 MCQ

Choose the incorrect statement about the two circles whose equations are given below :

x² + y² $-$ 10x $-$ 10y + 41 = 0 and

x² + y² $-$ 16x $-$ 10y + 80 = 0

A

Distance between two centres is the average of radii of both the circles.

B

Both circles pass through the centre of each other.

C

Circles have two intersection points.

D

Both circle's centers lie inside region of one another.

Question 76: Palette 76/145

JEE Mains 2021 MCQ

Let the lengths of intercepts on x-axis and y-axis made by the circle
x² + y² + ax + 2ay + c = 0, (a < 0) be 2${\sqrt 2 }$ and 2${\sqrt 5 }$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line x + 2y = 0, is equal to :

A

${\sqrt {10} }$

B

${\sqrt {6} }$

C

${\sqrt {11} }$

D

${\sqrt {7} }$

Question 77: Palette 77/145

JEE Mains 2021 MCQ

Let A(1, 4) and B(1, $-$5) be two points. Let P be a point on the circle
(x $-$ 1)² + (y $-$ 1)² = 1 such that (PA)² + (PB)² have maximum value, then the points, P, A and B lie on :

A

a straight line

B

an ellipse

C

a parabola

D

a hyperbola

Question 78: Palette 78/145

JEE Mains 2021 MCQ

If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x² + y² = 1 is a circle of radius r, then r is equal to :

A

${1 \over 4}$

B

${1 \over 2}$

C

1

D

${1 \over 3}$

Question 79: Palette 79/145

JEE Mains 2021 MCQ

In the circle given below, let OA = 1 unit, OB = 13 unit and PQ $ \bot $ OB. Then, the area of the triangle PQB (in square units) is :

A

24$\sqrt 2 $

B

24$\sqrt 3 $

C

26$\sqrt 2 $

D

26$\sqrt 3 $

Question 80: Palette 80/145

JEE Mains 2020 MCQ

If the length of the chord of the circle,
x² + y² = r² (r > 0) along the line, y – 2x = 3 is r,
then r² is equal to :

A

${9 \over 5}$

B

${{24} \over 5}$

C

${{12} \over 5}$

D

12

Question 81: Palette 81/145

JEE Mains 2020 MCQ

The circle passing through the intersection of the circles,
x² + y² – 6x = 0 and x² + y² – 4y = 0, having its centre on
the line, 2x – 3y + 12 = 0, also passes through the point :

A

(–3, 1)

B

(1, –3)

C

(–1, 3)

D

(–3, 6)

Question 82: Palette 82/145

JEE Mains 2020 MCQ

A circle touches the y-axis at the point (0, 4) and passes through the point (2, 0). Which of the following lines is not a tangent to this circle?

A

3x – 4y – 24 = 0

B

4x + 3y – 8 = 0

C

3x + 4y – 6 = 0

D

4x – 3y + 17 = 0

Question 83: Palette 83/145

JEE Mains 2020 MCQ

If a line, y = mx + c is a tangent to the circle, (x – 3)² + y² = 1 and it is perpendicular to a line L₁, where L₁ is the tangent to the circle, x² + y² = 1 at the point $\left( {{1 \over {\sqrt 2 }},{1 \over {\sqrt 2 }}} \right)$, then :

A

c² + 6c + 7 = 0

B

c² - 7c + 6 = 0

C

c² – 6c + 7 = 0

D

c² + 7c + 6 = 0

Question 84: Palette 84/145

JEE Mains 2020 MCQ

Let the tangents drawn from the origin to the circle,
x² + y² - 8x - 4y + 16 = 0 touch it at the points A and B. The (AB)² is equal to :

A

${{56} \over 5}$

B

${{32} \over 5}$

C

${{52} \over 5}$

D

${{64} \over 5}$

Question 85: Palette 85/145

JEE Mains 2019 MCQ

A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the point :

A

(1, 5)

B

( 2, 3)

C

(3, 5)

D

(3, 10)

Question 86: Palette 86/145

JEE Mains 2019 MCQ

If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90^o, then the length (in cm) of their common chord is :

A

${{13} \over 5}$

B

${{60} \over {13}}$

C

${{120} \over {13}}$

D

${{13} \over 2}$

Question 87: Palette 87/145

JEE Mains 2019 MCQ

The locus of the centres of the circles, which touch the circle, x² + y² = 1 externally, also touch the y-axis and lie in the first quadrant, is :

A

$x = \sqrt {1 + 2y} ,y \ge 0$

B

$y = \sqrt {1 + 2x} ,x \ge 0$

C

$y = \sqrt {1 + 4x} ,x \ge 0$

D

$x = \sqrt {1 + 4y} ,y \ge 0$

Question 88: Palette 88/145

JEE Mains 2019 MCQ

If the circles x² + y² + 5Kx + 2y + K = 0 and 2(x² + y²) + 2Kx + 3y –1 = 0, (K$ \in $R), intersect at the points P and Q, then the line 4x + 5y – K = 0 passes through P and Q, for :

A

exactly two values of K

B

no value of K

C

exactly one value of K

D

infinitely many values of K

Question 89: Palette 89/145

JEE Mains 2019 MCQ

The line x = y touches a circle at the point (1,1). If the circle also passes through the point (1, – 3), then its radius is :

A

3

B

2

C

2$\sqrt 2 $

D

3$\sqrt 2 $

Question 90: Palette 90/145

JEE Mains 2019 MCQ

A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (–8, 5) and (6, 5), then the area of the rectangle (in sq. units) is :

A

72

B

84

C

56

D

98

Question 91: Palette 91/145

JEE Mains 2019 MCQ

The common tangent to the circles x ² + y² = 4 and x² + y² + 6x + 8y – 24 = 0 also passes through the point :

A

(6, –2)

B

(4, –2)

C

(–4, 6)

D

(–6, 4)

Question 92: Palette 92/145

JEE Mains 2019 MCQ

If a tangent to the circle x² + y² = 1 intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is :

A

x² + y² – 4x²y² = 0

B

x² + y² - 2xy = 0

C

x² + y² – 2x²y² = 0

D

x² + y² - 16x²y² = 0

Question 93: Palette 93/145

JEE Mains 2019 MCQ

The tangent and the normal lines at the point ( $\sqrt 3 $, 1) to the circle x² + y² = 4 and the x-axis form a triangle. The area of this triangle (in square units) is :

A

${4 \over {\sqrt 3 }}$

B

${1 \over {\sqrt 3 }}$

C

${2 \over {\sqrt 3 }}$

D

${1 \over {3 }}$

Question 94: Palette 94/145

JEE Mains 2019 MCQ

The sum of the squares of the lengths of the chords intercepted on the circle, x² + y² = 16, by the lines, x + y = n, n $ \in $ N, where N is the set of all natural numbers, is :

A

210

B

160

C

320

D

105

Question 95: Palette 95/145

JEE Mains 2019 MCQ

If a circle of radius R passes through the origin O and intersects the coordinates axes at A and B, then the locus of the foot of perpendicular from O on AB is :

A

(x² + y²)² = 4R²x²y²

B

(x² + y²) (x + y) = R²xy

C

(x² + y²)² = 4Rx²y²

D

(x² + y²)³ = 4R²x²y²

Question 96: Palette 96/145

JEE Mains 2019 MCQ

If a variable line, 3x + 4y – $\lambda $ = 0 is such that the two circles x² + y² – 2x – 2y + 1 = 0 and x² + y² – 18x – 2y + 78 = 0 are on its opposite sides, then the set of all values of $\lambda $ is the interval :

A

(23, 31)

B

(2, 17)

C

[13, 23]

D

[12, 21]

Question 97: Palette 97/145

JEE Mains 2019 MCQ

Let C₁ and C₂ be the centres of the circles x² + y² – 2x – 2y – 2 = 0 and x² + y² – 6x – 6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral PC₁QC₂ is :

A

4

B

6

C

9

D

8

Question 98: Palette 98/145

JEE Mains 2019 MCQ

Two circles with equal radii are intersecting at the points (0, 1) and (0, –1). The tangent at the point (0, 1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is :

A

$2\sqrt 2 $

B

$\sqrt 2 $

C

2

D

1

Question 99: Palette 99/145

JEE Mains 2019 MCQ

A square is inscribed in the circle x² + y² – 6x + 8y – 103 = 0 with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is :

A

$\sqrt {137} $

B

6

C

$\sqrt {41} $

D

13

Question 100: Palette 100/145

JEE Mains 2019 MCQ

The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is :

A

$4\sqrt 5 $

B

${{\sqrt 5 } \over 2}$

C

$2\sqrt 5 $

D

${{\sqrt 5 } \over 4}$

Question 101: Palette 101/145

JEE Mains 2019 MCQ

If the area of an equilateral triangle inscribed in the circle x² + y² + 10x + 12y + c = 0 is $27\sqrt 3 $ sq units then c is equal to :

A

20

B

25

C

$-$ 25

D

13

Question 102: Palette 102/145

JEE Mains 2019 MCQ

If a circle C passing through the point (4, 0) touches the circle x² + y² + 4x – 6y = 12 externally at the point (1, – 1), then the radius of C is :

A

5

B

2$\sqrt {5} $

C

4

D

$\sqrt {37} $

Question 103: Palette 103/145

JEE Mains 2019 MCQ

If the circles

x² + y² $-$ 16x $-$ 20y + 164 = r²  

and  (x $-$ 4)² + (y $-$ 7)² = 36

intersect at two distinct points, then :

A

r > 11

B

0 < r < 1

C

r = 11

D

1 < r < 11

Question 104: Palette 104/145

JEE Mains 2019 MCQ

Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis as a common tangent, then :

A

a, b, c are in A.P.

B

$\sqrt a ,\sqrt b ,\sqrt c $ are in A.P

C

${1 \over {\sqrt b }} + {1 \over {\sqrt c }}$ = ${1 \over {\sqrt a }}$

D

${1 \over {\sqrt b }} = {1 \over {\sqrt a }} + {1 \over {\sqrt c }}$

Question 105: Palette 105/145

JEE Mains 2018 MCQ

If a circle C, whose radius is 3, touches externally the circle,
${x^2} + {y^2} + 2x - 4y - 4 = 0$ at the point (2, 2), then the length of the intercept cut by this circle C, on the x-axis is equal to :

A

$2\sqrt 5 $

B

$3\sqrt 2 $

C

$\sqrt 5 $

D

$2\sqrt 3 $

Question 106: Palette 106/145

JEE Mains 2018 MCQ

If the tangent at (1, 7) to the curve x² = y - 6

touches the circle x² + y² + 16x + 12y + c = 0, then the value of c is :

A

95

B

195

C

185

D

85

Question 107: Palette 107/145

JEE Mains 2018 MCQ

The tangent to the circle C₁ : x² + y² $-$ 2x $-$ 1 = 0 at the point (2, 1) cuts off a chord of length 4 from a circle C₂ whose center is (3, $-$2). The radius of C₂ is :

A

2

B

$\sqrt 2 $

C

3

D

$\sqrt 6 $

Question 108: Palette 108/145

JEE Mains 2018 MCQ

A circle passes through the points (2, 3) and (4, 5). If its centre lies on the line, $y - 4x + 3 = 0,$ then its radius is equal to :

A

2

B

$\sqrt 5 $

C

$\sqrt 2 $

D

1

Question 109: Palette 109/145

JEE Mains 2017 MCQ

The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60^o. If the area of the quadrilateral is $4\sqrt 3 $, then the perimeter of the quadrilateral is :

A

12.5

B

13.2

C

12

D

13

Question 110: Palette 110/145

JEE Mains 2017 MCQ

A line drawn through the point P(4, 7) cuts the circle x² + y² = 9 at the points A and B. Then PA⋅PB is equal to :

A

53

B

56

C

74

D

65

Question 111: Palette 111/145

JEE Mains 2017 MCQ

If two parallel chords of a circle, having diameter 4units, lie on the opposite sides of the center and subtend angles ${\cos ^{ - 1}}\left( {{1 \over 7}} \right)$ and sec^$-$1 (7) at the center respectivey, then the distance between these chords, is :

A

${4 \over {\sqrt 7 }}$

B

${8 \over {\sqrt 7 }}$

C

${8 \over 7}$

D

${16 \over 7}$

Question 112: Palette 112/145

JEE Mains 2017 MCQ

If a point P has co-ordinates (0, $-$2) and Q is any point on the circle, x² + y² $-$ 5x $-$ y + 5 = 0, then the maximum value of (PQ)² is :

A

${{25 + \sqrt 6 } \over 2}$

B

14 + $5\sqrt 3 $

C

${{47 + 10\sqrt 6 } \over 2}$

D

8 + 5$\sqrt 3 $

Question 113: Palette 113/145

JEE Mains 2017 MCQ

The radius of a circle, having minimum area, which touches the curve y = 4 – x² and the lines, y = |x| is :

A

$2\left( {\sqrt 2 - 1} \right)$

B

$4\left( {\sqrt 2 - 1} \right)$

C

$4\left( {\sqrt 2 + 1} \right)$

D

$2\left( {\sqrt 2 + 1} \right)$

Question 114: Palette 114/145

JEE Mains 2016 MCQ

Equation of the tangent to the circle, at the point (1, −1), whose centre is the point of intersection of the straight lines x − y = 1 and 2x + y = 3 is :

A

4x + y − 3 = 0

B

x + 4y + 3 = 0

C

3x − y − 4 = 0

D

x − 3y − 4 = 0

Question 115: Palette 115/145

JEE Mains 2016 MCQ

A circle passes through (−2, 4) and touches the y-axis at (0, 2). Which one of the following equations can represent a diameter of this circle?

A

4x + 5y − 6 = 0

B

2x − 3y + 10 = 0

C

3x + 4y − 3 = 0

D

5x + 2y + 4 = 0

Question 116: Palette 116/145

JEE Mains 2016 MCQ

If one of the diameters of the circle, given by the equation, ${x^2} + {y^2} - 4x + 6y - 12 = 0,$ is a chord of a circle $S$, whose centre is at $(-3, 2)$, then the radius of $S$ is :

A

$5$

B

$10$

C

$5\sqrt 2 $

D

$5\sqrt 3 $

Question 117: Palette 117/145

JEE Mains 2016 MCQ

The centres of those circles which touch the circle, ${x^2} + {y^2} - 8x - 8y - 4 = 0$, externally and also touch the $x$-axis, lie on :

A

a circle

B

an ellipse which is not a circle

C

a hyperbola

D

a parabola

Question 118: Palette 118/145

JEE Mains 2015 MCQ

Locus of the image of the point $(2, 3)$ in the line $\left( {2x - 3y + 4} \right) + k\left( {x - 2y + 3} \right) = 0,\,k \in R,$ is a :

A

circle of radius $\sqrt 2 $.

B

circle of radius $\sqrt 3 $.

C

straight line parallel to $x$-axis

D

straight line parallel to $y$-axis

Question 119: Palette 119/145

JEE Mains 2015 MCQ

The number of common tangents to the circles ${x^2} + {y^2} - 4x - 6x - 12 = 0$ and ${x^2} + {y^2} + 6x + 18y + 26 = 0,$ is :

A

$3$

B

$4$

C

$1$

D

$2$

Question 120: Palette 120/145

JEE Mains 2014 MCQ

Let $C$ be the circle with centre at $(1, 1)$ and radius $=$ $1$. If $T$ is the circle centred at $(0, y)$, passing through origin and touching the circle $C$ externally, then the radius of $T$ is equal to :

A

${1 \over 2}$

B

${1 \over 4}$

C

${{\sqrt 3 } \over {\sqrt 2 }}$

D

${{\sqrt 3 } \over 2}$

Question 121: Palette 121/145

JEE Mains 2013 MCQ

The circle passing through $(1, -2)$ and touching the axis of $x$ at $(3, 0)$ also passes through the point :

A

$\left( { - 5,\,2} \right)$

B

$\left( { 2,\,-5} \right)$

C

$\left( { 5,\,-2} \right)$

D

$\left( { - 2,\,5} \right)$

Question 122: Palette 122/145

JEE Mains 2012 MCQ

The length of the diameter of the circle which touches the $x$-axis at the point $(1, 0)$ and passes through the point $(2, 3)$ is :

A

${{10} \over 3}$

B

${{3} \over 5}$

C

${{6} \over 5}$

D

${{5} \over 3}$

Question 123: Palette 123/145

JEE Mains 2011 MCQ

The two circles x² + y² = ax, and x² + y² = c² (c > 0) touch each other if :

A

| a | = c

B

a = 2c

C

| a | = 2c

D

2 | a | = c

Question 124: Palette 124/145

JEE Mains 2010 MCQ

The circle ${x^2} + {y^2} = 4x + 8y + 5$ intersects the line $3x - 4y = m$ at two distinct points if :

A

$ - 35 < m < 15$

B

$ 15 < m < 65$

C

$ 35 < m < 85$

D

$ - 85 < m < -35$

Question 125: Palette 125/145

JEE Mains 2009 MCQ

Three distinct points A, B and C are given in the 2 -dimensional coordinates plane such that the ratio of the distance of any one of them from the point $(1, 0)$ to the distance from the point $(-1, 0)$ is equal to ${1 \over 3}$. Then the circumcentre of the triangle ABC is at the point :

A

$\left( {{5 \over 4},0} \right)$

B

$\left( {{5 \over 2},0} \right)$

C

$\left( {{5 \over 3},0} \right)$

D

$\left( {0,0} \right)$

Question 126: Palette 126/145

JEE Mains 2009 MCQ

If $P$ and $Q$ are the points of intersection of the circles
${x^2} + {y^2} + 3x + 7y + 2p - 5 = 0$ and ${x^2} + {y^2} + 2x + 2y - {p^2} = 0$ then there is a circle passing through $P,Q $ and $(1, 1)$ for :

A

all except one value of $p$

B

all except two values of $p$

C

exactly one value of $p$

D

all values of $p$

Question 127: Palette 127/145

JEE Mains 2008 MCQ

The differential equation of the family of circles with fixed radius $5$ units and centre on the line $y = 2$ is :

A

$\left( {x - 2} \right){y^2} = 25 - {\left( {y - 2} \right)^2}$

B

$\left( {y - 2} \right){y^2} = 25 - {\left( {y - 2} \right)^2}$

C

${\left( {y - 2} \right)^2}{y^2} = 25 - {\left( {y - 2} \right)^2}$

D

${\left( {x - 2} \right)^2}{y^2} = 25 - {\left( {y - 2} \right)^2}$

Question 128: Palette 128/145

JEE Mains 2008 MCQ

The point diametrically opposite to the point $P(1, 0)$ on the circle ${x^2} + {y^2} + 2x + 4y - 3 = 0$ is :

A

$(3, -4)$

B

$(-3, 4)$

C

$(-3, -4)$

D

$(3, 4)$

Question 129: Palette 129/145

JEE Mains 2007 MCQ

Consider a family of circles which are passing through the point $(-1, 1)$ and are tangent to $x$-axis. If $(h, k)$ are the coordinate of the centre of the circles, then the set of values of $k$ is given by the interval :

A

$ - {1 \over 2} \le k \le {1 \over 2}$

B

$k \le {1 \over 2}$

C

$0 \le k \le {1 \over 2}$

D

$k \ge {1 \over 2}$

Question 130: Palette 130/145

JEE Mains 2006 MCQ

If the lines $3x - 4y - 7 = 0$ and $2x - 3y - 5 = 0$ are two diameters of a circle of area $49\pi $ square units, the equation of the circle is :

A

$\,{x^2} + {y^2} + 2x\, - 2y - 47 = 0\,$

B

$\,{x^2} + {y^2} + 2x\, - 2y - 62 = 0\,$

C

${x^2} + {y^2} - 2x\, + 2y - 62 = 0$

D

${x^2} + {y^2} - 2x\, + 2y - 47 = 0$

Question 131: Palette 131/145

JEE Mains 2006 MCQ

Let $C$ be the circle with centre $(0, 0)$ and radius $3$ units. The equation of the locus of the mid points of the chords of the circle $C$ that subtend an angle of ${{2\pi } \over 3}$ at its center is :

A

${x^2} + {y^2} = {3 \over 2}$

B

${x^2} + {y^2} = 1$

C

${x^2} + {y^2} = {{27} \over 4}$

D

${x^2} + {y^2} = {{9} \over 4}$

Question 132: Palette 132/145

JEE Mains 2005 MCQ

If the circles ${x^2}\, + \,{y^2} + \,2ax\, + \,cy\, + a\,\, = 0$ and ${x^2}\, + \,{y^2} - \,3ax\, + \,dy\, - 1\,\, = 0$ intersect in two ditinct points P and Q then the line 5x + by - a = 0 passes through P and Q for :

A

exactly one value of a

B

no value of a

C

infinitely many values of a

D

exactly two values of a

Question 133: Palette 133/145

JEE Mains 2005 MCQ

If the pair of lines $a{x^2} + 2\left( {a + b} \right)xy + b{y^2} = 0$ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then :

A

$3{a^2} - 10ab + 3{b^2} = 0$

B

$3{a^2} - 2ab + 3{b^2} = 0$

C

$3{a^2} + 10ab + 3{b^2} = 0$

D

$3{a^2} + 2ab + 3{b^2} = 0$

Question 134: Palette 134/145

JEE Mains 2005 MCQ

A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is :

A

an ellipse

B

a circle

C

a hyperbola

D

a parabola

Question 135: Palette 135/145

JEE Mains 2005 MCQ

If a circle passes through the point (a, b) and cuts the circle ${x^2}\, + \,{y^2} = {p^2}$ orthogonally, then the equation of the locus of its centre is :

A

${x^2}\, + \,{y^2} - \,3ax\, - \,4\,by\,\, + \,({a^2}\, + \,{b^2} - {p^2}) = 0$

B

$2ax\, + \,\,2\,by\,\, - \,({a^2}\, - \,{b^2} + {p^2}) = 0$

C

${x^2}\, + \,{y^2} - \,2ax\, - \,\,3\,by\,\, + \,({a^2}\, - \,{b^2} - {p^2}) = 0$

D

$2ax\, + \,\,2\,by\,\, - \,({a^2}\, + \,{b^2} + {p^2}) = 0$

Question 136: Palette 136/145

JEE Mains 2004 MCQ

A variable circle passes through the fixed point A (p, q) and touches x-axis. The locus of the other end of the diameter through A is :

A

${(y\, - \,q)^2} = \,4\,px$

B

${(x\, - \,q)^2} = \,4\,py$

C

${(y\, - \,p)^2} = \,4\,qx$

D

${(x\, - \,p)^2} = \,4\,qy$

Question 137: Palette 137/145

JEE Mains 2004 MCQ

Intercept on the line y = x by the circle ${x^2}\, + \,{y^2} - 2x = 0$ is AB. Equation of the circle on AB as a diameter is :

A

$\,{x^2}\, + \,{y^2} + \,x\, - \,y\,\, = 0$

B

$\,{x^2}\, + \,{y^2} - \,x\, + \,y\,\, = 0$

C

$\,{x^2}\, + \,{y^2} + \,x\, + \,y\,\, = 0$

D

$\,{x^2}\, + \,{y^2} - \,x\, - \,y\,\, = 0$

Question 138: Palette 138/145

JEE Mains 2004 MCQ

If a circle passes through the point (a, b) and cuts the circle ${x^2}\, + \,{y^2} = 4$ orthogonally, then the locus of its centre is :

A

$2ax\, - 2by\, - ({a^2}\, + \,{b^2} + 4) = 0$

B

$2ax\, + 2by\, - ({a^2}\, + \,{b^2} + 4) = 0$

C

$2ax\, - 2by\, + ({a^2}\, + \,{b^2} + 4) = 0$

D

$2ax\, + 2by\, + ({a^2}\, + \,{b^2} + 4) = 0$

Question 139: Palette 139/145

JEE Mains 2004 MCQ

If the lines 2x + 3y + 1 + 0 and 3x - y - 4 = 0 lie along diameter of a circle of circumference $10\,\pi $, then the equation of the circle is :

A

${x^2}\, + \,{y^2} + \,2x\, - \,2y - \,23\,\, = 0$

B

${x^2}\, + \,{y^2} - \,2x\, - \,2y - \,23\,\, = 0$

C

${x^2}\, + \,{y^2} + \,2x\, + \,2y - \,23\,\, = 0$

D

${x^2}\, + \,{y^2} - \,2x\, + \,2y - \,23\,\, = 0$

Question 140: Palette 140/145

JEE Mains 2003 MCQ

The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle having area as 154 sq. units. Then the equation of the circle is :

A

${x^2}\, + \,{y^2} - \,2x\, + \,2y\,\, = \,62$

B

${x^2}\, + \,{y^2} + \,2x\, - \,2y\,\, = \,62$

C

${x^2}\, + \,{y^2} + \,2x\, - \,2y\,\, = \,47$

D

${x^2}\, + \,{y^2} - \,2x\, + \,2y\,\, = \,47$

Question 141: Palette 141/145

JEE Mains 2003 MCQ

If the two circles ${(x - 1)^2}\, + \,{(y - 3)^2} = \,{r^2}$ and $\,{x^2}\, + \,{y^2} - \,8x\, + \,2y\, + \,\,8\,\, = 0$ intersect in two distinct point, then :

A

$r > 2$

B

$2 < r < 8$

C

$r < 2$

D

$r = 2.$

Question 142: Palette 142/145

JEE Mains 2002 MCQ

The centres of a set of circles, each of radius 3, lie on the circle ${x^2}\, + \,{y^2} = 25$. The locus of any point in the set is :

A

$4\, \le \,\,{x^2}\, + \,{y^2}\, \le \,\,64$

B

${x^2}\, + \,{y^2}\, \le \,\,25$

C

${x^2}\, + \,{y^2}\, \ge \,\,25$

D

$3\, \le \,\,{x^2}\, + \,{y^2}\, \le \,\,9$

Question 143: Palette 143/145

JEE Mains 2002 MCQ

If the chord y = mx + 1 of the circle ${x^2}\, + \,{y^2} = 1$ subtends an angle of measure ${45^ \circ }$ at the major segment of the circle then value of m is :

A

$2\, \pm \,\sqrt 2 \,\,$

B

$ - \,2\, \pm \,\sqrt 2 \,$

C

$- 1\, \pm \,\sqrt 2 \,\,$

D

none of these

Question 144: Palette 144/145

JEE Mains 2002 MCQ

The equation of a circle with origin as a center and passing through an equilateral triangle whose median is of length $3$$a$ is :

A

${x^2}\, + \,{y^2} = 9{a^2}$

B

${x^2}\, + \,{y^2} = 16{a^2}$

C

${x^2}\, + \,{y^2} = 4{a^2}$

D

${x^2}\, + \,{y^2} = {a^2}$

Question 145: Palette 145/145

JEE Mains 2002 MCQ

The centre of the circle passing through (0, 0) and (1, 0) and touching the circle ${x^2}\, + \,{y^2} = 9$ is :

A

$\left( {{1 \over 2},\,{1 \over 2}} \right)$

B

$\left( {{1 \over 2},\, - \,\sqrt 2 } \right)$

C

$\left( {{3 \over 2},\,{1 \over 2}} \right)$

D

$\left( {{1 \over 2},\,{3 \over 2}} \right)$