iCON Education HYD, 79930 92826, 73309 72826JEE Main 2018 (Online) 15th April Evening Slot
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 $
Correct Answer: D
Explanation:
Here, equation of tangent on C1 at (2, 1) is :
2x + y $-$ (x + 2) $-$1 = 0
Or x + y = 3
If it cuts off the chord of the circle C2 then the equation of the chord is :
x + y = 3
$\therefore\,\,\,$ distance of the chord from (3, $-$ 2) is :
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2017 (Online) 9th April Morning Slot
The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60o. If the area of the quadrilateral is $4\sqrt 3 $, then the perimeter
of the quadrilateral is :
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2017 (Online) 8th April Morning Slot
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 :
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2016 (Online) 10th April Morning Slot
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
Correct Answer: B
Explanation:
Point of intersection of lines
x $-$ y = 1 and 2x + y = 3 is $\left( {{4 \over 3},{1 \over 3}} \right)$
Equation of tangent y + 1 = $-$ ${1 \over 4}$ (x $-$ 1)
4y + 4 = $-$ x + 1
x + 4y + 3 = 0
2016
JEE Mains
MCQ
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2016 (Online) 9th April Morning Slot
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
Correct Answer: B
Explanation:
EF = perpendicular bisector of chord AB
BG = perpendicular to y-axis
Here C = center of the circle
mid-point of chord AB, D = ($-$ 1, 3)
slope of AB = ${{4 - 2} \over { - 2 - 0}}$ = $-$ 1
$ \because $ EF $ \bot $ AB
$ \therefore $ Slope of EF = 1
Equation of EF, y $-$ 3 = 1 (x + 1)
$ \Rightarrow $ y = x + 4 . . . .(i)
Equation of BG
y = 2 . . . . (ii)
From equations (i) and (ii)
x = $-$ 2, y = 2
since C be the point of intersection of EF and BG, therefore center, C = ($-$ 2, 2)
Now coordinates of center C satiesfy the equation
2x $-$ 3y + 10 = 0
Hence 2x $-$ 3y + 10 = 0 is the equation of the diameter
2016
JEE Mains
MCQ
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2016 (Offline)
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 $
Correct Answer: D
Explanation:
Center of $S$ : $O(-3, 2)$ center of given circle $A(2, -3)$
$ \Rightarrow OA = 5\sqrt 2 $
Also $AB=5$ (as $AB=r$ of the given circle)
$ \Rightarrow $ Using pythagoras theorem in $\Delta OAB$
$r = 5\sqrt 3 $
2016
JEE Mains
MCQ
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2016 (Offline)
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 :
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2014 (Offline)
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.
${{\sqrt 3 } \over 2}$
Correct Answer: B
Explanation:
Equation of circle $C \equiv {\left( {x - 1} \right)^2} + {\left( {y - 1} \right)^2} = 1$
Radius of $T = \left| y \right|$
$T$ touches $C$ externally
therefore,
Distance between the centers $=$ sum of their radii
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 :
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 :
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.
$k \ge {1 \over 2}$
Correct Answer: D
Explanation:
Equation of circle whose center is $\left( {h,k} \right)$
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.
${x^2} + {y^2} = {{9} \over 4}$
Correct Answer: D
Explanation:
Let $M\left( {h,k} \right)$ be the mid point of chord $AB$ where
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
Correct Answer: B
Explanation:
${s_1} = {x^2} + {y^2} + 2ax + cy + a = 0$
${s_2} = {x^2} + {y^2} - 3ax + dy - 1 = 0$
Equation of common chord of circles ${s_1}$ and ${s_2}$ is
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.
$3{a^2} + 2ab + 3{b^2} = 0$
Correct Answer: D
Explanation:
As per question area of one sector $=3$ area of another sector
$ \Rightarrow $ at center by one sector $ = 3 \times $ angle at center by another sector
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 :
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.
$r = 2.$
Correct Answer: B
Explanation:
$\left| {{r_1} - {r_2}} \right| < {C_1}{C_2}$ for intersection
$ \Rightarrow r - 3 < 5 \Rightarrow r < 8\,\,\,\,\,\,\,\,\,...\left( 1 \right)$
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
Correct Answer: C
Explanation:
Equation of circle ${x^2} + {y^2} = 1 = {\left( 1 \right)^2}$
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2026 (Online) 21st January Evening Shift
If $P$ is a point on the circle $x^2+y^2=4, Q$ is a point on the straight line $5 x+y+2=0$ and $x-y+1=0$ is the perpendicular bisector of PQ , then 13 times the sum of abscissa of all such points P is $\_\_\_\_$ .
Correct Answer: 2
Explanation:
$P\left(x_1 y_1\right)$ and point $Q\left(x_2, y_2\right)$
Mid point of $\mathrm{PQ} M=\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
Substitute M into $x-y+1=0$
$ x_1+x_2-y_1-y_2+2=0 . .(i) $
Slope of PQ is Perpendicular to slope of bisector line So, slope of $P Q=-1$
$ y_2=x_1-x_2+y_1 \ldots . .(i i) $
$Q\left(x_2, y_2\right)$ lie on $5 x+y+2=0$
So, $5 x_2+y_2+2=0 \ldots \ldots$. (iii)
Substitute (iii) in (i)
$ x_2=\frac{-x_1-y_1-2}{4} \ldots .(i v) $
Substitute (iii) in (ii)
$ x_2=y_1-1 \ldots . .(v) $
From (iv) and (v)
$ x_1=2-5 y $
$\left(x_1, y_1\right)$ lie on circle
$ x_1^2+y_1^2=4 $
Pt $x_1=2-5 y_1$
$ y_1=0,-\frac{10}{13} $
So, $x_1=2, \frac{-24}{13}$
So, $2+\left(-\frac{24}{13}\right)=\frac{2}{13}$
So, $13 \times \frac{2}{13}=2$
2025
JEE Mains
Numerical
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2025 (Online) 4th April Morning Shift
Let $C$ be the circle $x^2+(y-1)^2=2, E_1$ and $E_2$ be two ellipses whose centres lie at the origin and major axes lie on x -axis and y -axis respectively. Let the straight line $x+y=3$ touch the curves $C, E_1$ and $E_2$ at $P\left(x_1, y_1\right), Q\left(x_2, y_2\right)$ and $R\left(x_3, y_3\right)$ respectively. Given that $P$ is the mid point of the line segment $Q R$ and $P Q=\frac{2 \sqrt{2}}{3}$, the value of $9\left(x_1 y_1+x_2 y_2+x_3 y_3\right)$ is equal to _______.
Correct Answer: 46
Explanation:
Solving the line $x+y=3$, and the circle $x^2+$ $(y-1)^2=2$
$\begin{aligned}
& x_1 y_1+x_2 y_2+x_3 y_3=2+a b+(2-a)(4-b) \\
& 9\left(x_1 y_1+x_2 y_2+x_3 y_3\right)=9(10+2 a b-2 b-4 a) \\
& =90+18 a b-18 b-36 a=46
\end{aligned}$
2025
JEE Mains
Numerical
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2025 (Online) 2nd April Morning Shift
The absolute difference between the squares of the radii of the two circles passing through the point $(-9,4)$ and touching the lines $x+y=3$ and $x-y=3$, is equal to ________ .
Correct Answer: 768
Explanation:
$\because x+y=3$ and $x-y=3$ are tangents
$\therefore \quad$ Both circle centre will lie on $x$-axis
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2025 (Online) 23rd January Morning Shift
Let the circle $C$ touch the line $x-y+1=0$, have the centre on the positive $x$-axis, and cut off a chord of length $\frac{4}{\sqrt{13}}$ along the line $-3 x+2 y=1$. Let H be the hyperbola $\frac{x^2}{\alpha^2}-\frac{y^2}{\beta^2}=1$, whose one of the foci is the centre of $C$ and the length of the transverse axis is the diameter of $C$. Then $2 \alpha^2+3 \beta^2$ is equal to ________.
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2024 (Online) 9th April Morning Shift
Let the centre of a circle, passing through the points $(0,0),(1,0)$ and touching the circle $x^2+y^2=9$, be $(h, k)$. Then for all possible values of the coordinates of the centre $(h, k), 4\left(h^2+k^2\right)$ is equal to __________.
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2024 (Online) 30th January Evening Shift
Consider two circles $C_1: x^2+y^2=25$ and $C_2:(x-\alpha)^2+y^2=16$, where $\alpha \in(5,9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_1$ and $C_2$ be $\sin ^{-1}\left(\frac{\sqrt{63}}{8}\right)$. If the length of common chord of $C_1$ and $C_2$ is $\beta$, then the value of $(\alpha \beta)^2$ equals _______.
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2024 (Online) 29th January Morning Shift
Equations of two diameters of a circle are $2 x-3 y=5$ and $3 x-4 y=7$. The line joining the points $\left(-\frac{22}{7},-4\right)$ and $\left(-\frac{1}{7}, 3\right)$ intersects the circle at only one point $P(\alpha, \beta)$. Then, $17 \beta-\alpha$ is equal to _________.
Correct Answer: 2
Explanation:
Centre of circle is $(1,-1)$
Equation of $A B$ is $7 x-3 y+10=0 \ldots$ (i)
Equation of $\mathrm{CP}$ is $3 x+7 y+4=0 \ldots$ (ii)
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2024 (Online) 27th January Evening Shift
Consider a circle $(x-\alpha)^2+(y-\beta)^2=50$, where $\alpha, \beta>0$. If the circle touches the line $y+x=0$ at the point $P$, whose distance from the origin is $4 \sqrt{2}$, then $(\alpha+\beta)^2$ is equal to __________.
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2023 (Online) 12th April Morning Shift
Two circles in the first quadrant of radii $r_{1}$ and $r_{2}$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $x+y=2$. Then $r_{1}^{2}+r_{2}^{2}-r_{1} r_{2}$ is equal to ___________.
Correct Answer: 7
Explanation:
$
\begin{aligned}
& \text { Circle }(x-a)^2+(y-a)^2=a^2 \\\\
& x^2+y^2-2 a x-2 a y+a^2=0 \\\\
& \text { intercept }=2 \\\\
& \Rightarrow 2 \sqrt{a^2-d^2}=2
\end{aligned}
$
Where $\mathrm{d}=$ perpendicular distance of centre from line $x+y=2$
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2023 (Online) 8th April Morning Shift
Consider a circle $C_{1}: x^{2}+y^{2}-4 x-2 y=\alpha-5$. Let its mirror image in the line $y=2 x+1$ be another circle $C_{2}: 5 x^{2}+5 y^{2}-10 f x-10 g y+36=0$. Let $r$ be the radius of $C_{2}$. Then $\alpha+r$ is equal to _________.
