Question 1: Palette 1/53

JEE Advanced 2026 MCQ

Let $P$ be the point on the parabola $y = x^2$ such that the slope of the tangent to the parabola at the point $P$ is $4$. Let $Q$ be the point in the first quadrant lying on the circle $x^2 + y^2 = 2$ such that the slope of the tangent to the circle at the point $Q$ is $-1$. Let $R$ be the point in the first quadrant lying on the ellipse $x^2 + 4y^2 = 8$ such that the slope of the tangent to the ellipse at the point $R$ is $-\frac{1}{2}$. Then the radius of the circle passing through the points $P, Q$ and $R$ is

A

$\sqrt{10}$

B

$\sqrt{5}$

C

$\sqrt{\dfrac{5}{2}}$

D

$2\sqrt{5}$

Question 2: Palette 2/53

JEE Advanced 2024 MCQ

Let the straight line $y=2 x$ touch a circle with center $(0, \alpha), \alpha>0$, and radius $r$ at a point $A_1$. Let $B_1$ be the point on the circle such that the line segment $A_1 B_1$ is a diameter of the circle. Let $\alpha+r=5+\sqrt{5}$.

Match each entry in List-I to the correct entry in List-II.

List-I	List-II
(P) $\alpha$ equals	(1) $(-2, 4)$
(Q) $r$ equals	(2) $\sqrt{5}$
(R) $A_1$ equals	(3) $(-2, 6)$
(S) $B_1$ equals	(4) $5$
	(5) $(2, 4)$

The correct option is

A

$(\mathrm{P}) \rightarrow(4) \quad(\mathrm{Q}) \rightarrow(2) \quad(\mathrm{R}) \rightarrow(1) \quad(\mathrm{S}) \rightarrow(3)$

B

$(\mathrm{P}) \rightarrow(2) \quad(\mathrm{Q}) \rightarrow(4) \quad(\mathrm{R}) \rightarrow(1) \quad(\mathrm{S}) \rightarrow(3)$

C

$(\mathrm{P}) \rightarrow(4) \quad(\mathrm{Q}) \rightarrow(2) \quad(\mathrm{R}) \rightarrow(5) \quad(\mathrm{S}) \rightarrow(3)$

D

$(\mathrm{P}) \rightarrow(2) \quad(\mathrm{Q}) \rightarrow(4) \quad(\mathrm{R}) \rightarrow(3) \quad(\mathrm{S}) \rightarrow(5)$

Question 3: Palette 3/53

JEE Advanced 2022 MSQ

Let $G$ be a circle of radius $R>0$. Let $G_{1}, G_{2}, \ldots, G_{n}$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_{1}, G_{2}, \ldots, G_{n}$ touches the circle $G$ externally. Also, for $i=1,2, \ldots, n-1$, the circle $G_{i}$ touches $G_{i+1}$ externally, and $G_{n}$ touches $G_{1}$ externally. Then, which of the following statements is/are TRUE?

A

If $n=4$, then $(\sqrt{2}-1) r < R$

B

If $n=5$, then $r < R$

C

If $n=8$, then $(\sqrt{2}-1) r < R$

D

If $n=12$, then $\sqrt{2}(\sqrt{3}+1) r > R$

Question 4: Palette 4/53

JEE Advanced 2021 MCQ

Consider M with $r = {{1025} \over {513}}$. Let k be the number of all those circles C_n that are inside M. Let l be the maximum possible number of circles among these k circles such that no two circles intersect. Then

A

k + 2l = 22

B

2k + l = 26

C

2k + 3l = 34

D

3k + 2l = 40

Question 5: Palette 5/53

JEE Advanced 2021 MCQ

Consider M with $r = {{({2^{199}} - 1)\sqrt 2 } \over {{2^{198}}}}$. The number of all those circles D_n that are inside M is

A

198

B

199

C

200

D

201

Question 6: Palette 6/53

JEE Advanced 2021 MCQ

Consider a triangle $\Delta$ whose two sides lie on the x-axis and the line x + y + 1 = 0. If the orthocenter of $\Delta$ is (1, 1), then the equation of the circle passing through the vertices of the triangle $\Delta$ is

A

x² + y² $-$ 3x + y = 0

B

x² + y² + x + 3y = 0

C

x² + y² + 2y $-$ 1 = 0

D

x² + y² + x + y = 0

Question 7: Palette 7/53

JEE Advanced 2019 MCQ

A line y = mx + 1 intersects the circle ${(x - 3)^2} + {(y + 2)^2}$ = 25 at the points P and Q. If the midpoint of the line segment PQ has x-coordinate $ - {3 \over 5}$, then which one of the following options is correct?

A

6 $ \le $ m < 8

B

$ - $3 $ \le $ m < $ - $1

C

4 $ \le $ m < 6

D

2 $ \le $ m < 4

Question 8: Palette 8/53

JEE Advanced 2018 MCQ

Let S be the circle in the XY-plane defined the equation x² + y² = 4.

Let E₁E₂ and F₁F₂ be the chords of S passing through the point P₀ (1, 1) and parallel to the X-axis and the Y-axis, respectively. Let G₁G₂ be the chord of S passing through P₀ and having slope$-$1. Let the tangents to S at E₁ and E₂ meet at E₃, then tangents to S at F₁ and F₂ meet at F₃, and the tangents to S at G₁ and G₂ meet at G₃. Then, the points E₃, F₃ and G₃ lie on the curve

A

x + y = 4

B

(x $-$ 4)² + (y $-$ 4)² = 16

C

(x $-$ 4)(y $-$ 4) = 4

D

xy = 4

Question 9: Palette 9/53

JEE Advanced 2016 MSQ

Let RS be the diameter of the circle ${x^2}\, + \,{y^2} = 1$, where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes through the point (s)

A

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

B

$\left( {{1 \over 4}\,,{1 \over 2}} \right)$

C

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

D

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

Question 10: Palette 10/53

JEE Advanced 2014 MSQ

A circle S passes through the point (0, 1) and is orthogonal to the circles ${(x - 1)^2}\, + \,{y^2} = 16\,\,and\,\,{x^2}\, + \,{y^2} = 1$. Then

A

radius of S is 8

B

radius of S is 7

C

centre of S is (- 7, 1)

D

centre of S is (- 8, 1)

Question 11: Palette 11/53

JEE Advanced 2013 MSQ

Circle (s) touching x-axis at a distance 3 from the origin and having an intercept of length $2\sqrt 7 $ on y-axis is (are)

A

${x^2}\, + \,{y^2}\, - \,6x\,\, + 8y\, + 9 = 0$

B

${x^2}\, + \,{y^2}\, - \,6x\,\, + 7y\, + 9 = 0$

C

${x^2}\, + \,{y^2}\, - \,6x\,\, - 8y\, + 9 = 0$

D

${x^2}\, + \,{y^2}\, - \,6x\,\,- 7y\, + 9 = 0$

Question 12: Palette 12/53

JEE Advanced 2012 MCQ

A tangent PT is drawn to the circle ${x^2}\, + {y^2} = 4$ at the point P $\left( {\sqrt 3 ,1} \right)$. A straight line L, perpendicular to PT is a tangent to the circle ${(x - 3)^2}$ + ${y^2}$ = 1.

A possible equation of L is

A

${x - \sqrt 3 \,y = 1}$

B

${x + \sqrt 3 \,y = 1}$

C

${x - \sqrt 3 \,y = -1}$

D

${x + \sqrt 3 \,y = 5}$

Question 13: Palette 13/53

JEE Advanced 2012 MCQ

A tangent PT is drawn to the circle ${x^2}\, + {y^2} = 4$ at the point P $\left( {\sqrt 3 ,1} \right)$. A straight line L, perpendicular to PT is a tangent to the circle ${(x - 3)^2}$ + ${y^2}$ = 1

A common tangent of the two circles is

A

x = 4

B

y = 2

C

${x + \sqrt 3 \,y = 4}$

D

${x +2 \sqrt 2 \,y = 6}$

Question 14: Palette 14/53

JEE Advanced 2012 MCQ

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x - 5y = 20 to the circle ${x^2}\, + \,{y^2} = 9$ is

A

$20\,({x^2}\, + \,{y^2}) - \,\,36x\,\, + \,\,45y = 0$

B

$20\,({x^2}\, + \,{y^2}) + \,\,36x\,\, - \,\,45y = 0$

C

$36\,({x^2}\, + \,{y^2}) - \,\,20x\,\, + \,\,45y = 0$

D

$36\,({x^2}\, + \,{y^2}) + \,\,20x\,\, - \,\,45y = 0$

Question 15: Palette 15/53

JEE Advanced 2011 MCQ

The circle passing through the point (-1, 0) and touching the y-axis at (0, 2) also passes through the point.

A

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

B

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

C

$\left( { - {3 \over 0},\,{5 \over 2}} \right)$

D

(- 4, 0)

Question 16: Palette 16/53

JEE Advanced 2009 MCQ

Tangents drawn from the point P (1, 8) to the circle
${x^2}\, + \,{y^2}\, - \,6x\, - 4y\, - 11 = 0$
touch the circle at the points A and B. The equation of the cirumcircle of the triangle PAB is

A

${x^2}\, + \,{y^2}\, + \,4x\,\, - 6y\, + 19 = 0$

B

${x^2}\, + \,{y^2}\, - \,4x\,\, - 10y\, + 19 = 0$

C

${x^2}\, + \,{y^2}\, - \,2x\,\, + 6y\, - 29 = 0$

D

${x^2}\, + \,{y^2}\, - \,6x\,\, - 4y\, + 19 = 0$

Question 17: Palette 17/53

JEE Advanced 2008 MCQ

Consider

$\,{L_1}:\,\,2x\,\, + \,\,3y\, + \,p\,\, - \,\,3 = 0$

$\,{L_2}:\,\,2x\,\, + \,\,3y\, + \,p\,\, + \,\,3 = 0$

where p is a real number, and $\,C:\,{x^2}\, + \,{y^2}\, + \,6x\, - 10y\, + \,30 = 0$

STATEMENT-1 : If line ${L_1}$ is a chord of circle C, then line ${L_2}$ is not always a diameter of circle C
and

STATEMENT-2 : If line ${L_1}$ is a diameter of circle C, then line ${L_2}$ is not a chord of circle C.

A

Statement-1 is True, Statement-2 is True; Statement-2 is a correct rexplanation for Statement-1

B

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct rexplanation for Statement-1

C

Statement-1 is True, Statement-2 is False

D

Statement-1 is False, Statement-2 is True

Question 18: Palette 18/53

JEE Advanced 2008 MCQ

Points E and F are given by

A

$\left( {{{\,\sqrt 3 } \over 2},\,{3 \over 2}} \right),\,\left( {\sqrt 3 ,\,0} \right)$

B

$\left( {{{\,\sqrt 3 } \over 2},\,{1 \over 2}} \right),\,\left( {\sqrt 3 ,\,0} \right)$

C

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

D

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

Question 19: Palette 19/53

JEE Advanced 2008 MCQ

Equations of the sides QR, RP are

A

$y = {2 \over {\sqrt 3 }}\,x + \,1,\,\,y = \, - {2 \over {\sqrt 3 }}\,x - 1$

B

$y = {1 \over {\sqrt 3 }}\,x,\,\,y = \,0$

C

$y = {{\sqrt 3 } \over 2}\,x + \,1,\,\,y = \, - {{\sqrt 3 } \over 2}\,x - 1$

D

$y = \sqrt 3 \,x,\,\,y = \,0$

Question 20: Palette 20/53

JEE Advanced 2008 MCQ

The equation of circle C is

A

${\left( {x\, - 2\sqrt 3 \,} \right)^2} + {(y - 1)^2} = 1$

B

${\left( {x\, - 2\sqrt 3 \,} \right)^2} + {(y + {1 \over 2})^2} = 1$

C

${\left( {x\, - \sqrt 3 \,} \right)^2} + {(y + 1)^2} = 1$

D

${\left( {x\, - \sqrt 3 \,} \right)^2} + {(y - 1)^2} = 1$

Question 21: Palette 21/53

JEE Advanced 2007 MCQ

Let $\mathrm{ABCD}$ be a quadrilateral with area 18 , with side $\mathrm{A B}$ parallel to the side $\mathrm{C D}$ and $\mathrm{A B}=2 \mathrm{CD}$. Let $\mathrm{AD}$ be perpendicular to $\mathrm{AB}$ and $\mathrm{CD}$. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is :

A

3

B

2

C

$\frac{3}{2}$

D

1

Question 22: Palette 22/53

JEE Advanced 2007 MCQ

Match the statements in Column I with the properties Column II.

	Column I		Column II
(A)	Two intersecting circles	(P)	have a common tangent
(B)	Two mutually external circles	(Q)	have a common normal
(C)	Two circles, one strictly inside the other	(R)	do not have a common tangent
(D)	Two branches of a hyperbola	(S)	do not have a common normal

A

$\mathrm{A-(p);B-(p),(q);C-(q),(r);D-(q)}$

B

$\mathrm{A-(p),(q);B-(q);C-(r);D-(q),(r)}$

C

$\mathrm{A-(q);B-(p),(q);C-(q),(r);D-(r)}$

D

$\mathrm{A-(p),(q);B-(p),(q);C-(q),(r);D-(q),(r)}$

Question 23: Palette 23/53

JEE Advanced 2007 MCQ

Tangents are drawn from the point (17, 7) to the circle $x^2+y^2=169$.

Statement 1 : The tangents are mutually perpendicular.

Statement 2 : The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is $x^2+y^2=338$

A

Statement 1 is True, Statement 2 is True, Statement 2 is a CORRECT explanation for Statement 1

B

Statement 1 is True, Statement 2 is True, Statement 2 is NOT a CORRECT explanation for Statement 1

C

Statement 1 is True, Statement 2 is False

D

Statement 1 is False, Statement 2 is True

Question 24: Palette 24/53

JEE Advanced 2006 MCQ

A circle touches the line $L$ and the circle $C_1$ externally such that both the circles are on the same side of the line, then the locus of center of the circle is:

A

ellipse

B

hyperbola

C

parabola

D

parts of straight line

Question 25: Palette 25/53

JEE Advanced 2006 MCQ

A line $M$ through $A$ is drawn parallel to $B D$. Point $S$ moves such that its distances from

the line BD and the vertex A are equal. If locus of S cuts M at $\mathrm{T}_2$ and $\mathrm{T}_3$ and AC at $\mathrm{T}_1$, then area of $\Delta T_1 T_2 T_3$ is :

A

$\frac{1}{2}$ sq. units

B

$\frac{2}{3}$ sq. units

C

1 sq. unit

D

2 sq. units

Question 26: Palette 26/53

JEE Advanced 2005 MCQ

A circle is given by ${x^2}\, + \,{(y\, - \,1\,)^2}\, = \,1$, another circle C touches it externally and also the x-axis, then thelocus of its centre is

A

$\{ (x,\,y):\,\,{x^2} = \,4y\} \, \cup \,\{ (x,\,y):\,\,y \le \,0\,\} $

B

$\{ (x,\,y):\,\,{x^2} + \,{(y\, - \,1)^2}\, = \,4\} \, \cup \,\{ (x,\,\,y):\,\,y \le \,0\,\} $

C

$\{ (x,\,y):\,\,{x^2} = \,y\} \, \cup \,\{ (0,\,\,y):\,\,y \le \,0\,\} $

D

$\{ (x,\,y):\,\,{x^2} = \,4y\} \, \cup \,\{ (0,\,\,y):\,\,y \le \,0\,\} $

Question 27: Palette 27/53

JEE Advanced 2005 MCQ

Circles with radii 3, 4 and 5 touch each other externally if P is the point of intersection of tangents to these circles at their points of contact. Find the distance of P from the point of contact.

A

5

B

$\sqrt3$

C

$\sqrt5$

D

3

Question 28: Palette 28/53

JEE Advanced 2004 MCQ

If one of the diameters of the circle ${x^2} + {y^2} - 2x - 6y + 6 = 0$ is a chord to the circle with centre (2, 1), then the radius of the circle is

A

${\sqrt 3 }$

B

${\sqrt 2 }$

C

3

D

2

Question 29: Palette 29/53

JEE Advanced 2003 MCQ

The centre of circle inscibed in square formed by the lines ${x^2} - 8x + 12 = 0\,\,and\,{y^2} - 14y + 45 = 0$, is

A

(4, 7)

B

(7, 4)

C

(9, 4)

D

(4, 9)

Question 30: Palette 30/53

JEE Advanced 2002 MCQ

If the tangent at the point P on the circle ${x^2} + {y^2} + 6x + 6y = 2$ meets a straight line 5x - 2y + 6 = 0 at a point Q on the y-axis, then the lenght of PQ is

A

4

B

${2\sqrt 5 }$

C

5

D

${3\sqrt 5 }$

Question 31: Palette 31/53

JEE Advanced 2002 MCQ

If $a > 2b > 0$ then the positive value of $m$ for which $y = mx - b\sqrt {1 + {m^2}} $ is a common tangent to ${x^2} + {y^2} = {b^2}$ and ${\left( {x - a} \right)^2} + {y^2} = {b^2}$ is

A

${{2b} \over {\sqrt {{a^2} - 4{b^2}} }}$

B

${{\sqrt {{a^2} - 4{b^2}} } \over {2b}}$

C

${{2b} \over {a - 2b}}$

D

${{b} \over {a - 2b}}$

Question 32: Palette 32/53

JEE Advanced 2001 MCQ

Let A B be a chord of the circle ${x^2} + {y^2} = {r^2}$ subtending a right angle at the centre. Then the locus of the centriod of the triangle PAB as P moves on the circle is

A

a parabola

B

a circle

C

an ellipse

D

a pair of straight lines

Question 33: Palette 33/53

JEE Advanced 2001 MCQ

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals

A

$\sqrt {PQ.\,RS} $

B

(PQ + RS) / 2

C

2 PQ. RS/(PQ + RS)

D

$\sqrt {\left( {P{Q^2} + \,R{S^2}} \right)} \,\,/2$

Question 34: Palette 34/53

JEE Advanced 2000 MCQ

If the circles ${x^2}\, + \,{y^2}\, + \,\,2x\, + \,2\,k\,y\,\, + \,6\,\, = \,\,0,\,\,{x^2}\, + \,\,{y^2}\, + \,2ky\, + \,k\, = \,0$ intersect orthogonally, then k is

A

2 or $ - {3 \over 2}$

B

- 2 or $ - {3 \over 2}$

C

2 or $ {3 \over 2}$

D

- 2 or $ {3 \over 2}$

Question 35: Palette 35/53

JEE Advanced 2000 MCQ

The triangle PQR is inscribed in the circle ${x^2}\, + \,\,{y^2} = \,25$. If Q and R have co-ordinates (3, 4) and ( - 4, 3) respectively, then $\angle \,Q\,P\,R$ is equal to

A

${\pi \over 2}$

B

${\pi \over 3}$

C

${\pi \over 4}$

D

${\pi \over 6}$

Question 36: Palette 36/53

JEE Advanced 1999 MCQ

If two distinct chords, drawn from the point (p, q) on the circle ${x^2}\, + \,{y^2} = \,px\, + \,qy\,\,(\,where\,pq\, \ne \,0)$ are bisected by the x - axis, then

A

${p^2}\, = \,\,{q^2}$

B

$\,{p^2}\, = \,\,8\,{q^2}$

C

${p^2}\, < \,\,8\,{q^2}$

D

${p^2}\, > \,\,8\,{q^2}$.

Question 37: Palette 37/53

JEE Advanced 1998 MCQ

The number of common tangents to the circles ${x^2}\, + \,{y^2} = 4$ and ${x^2}\, + \,{y^2}\, - 6x\, - 8y = 24$ is

A

0

B

1

C

3

D

4

Question 38: Palette 38/53

JEE Advanced 1998 MSQ

If the circle ${x^2}\, + \,{y^2} = \,{a^2}$ intersects the hyperbola $xy = {c^2}$ in four points $P\,({x_1},\,{y_1}),\,Q\,\,({x_2},\,{y_2}),\,\,R\,({x_3},\,{y_3}),\,S\,({x_4},\,{y_4}),$ then

A

${x_1}\, + \,{x_2} + \,{x_3}\, + \,{x_4}\, = 0$

B

${y_1}\, + \,{y_2} + \,{y_3}\, + \,{y_4}\, = 0$

C

${x_1}\,{x_2}\,{x_3}\,{x_4}\, = {c^4}$

D

${y_1}\,{y_2}\,{y_3}\,{y_4}\, = {c^4}$

Question 39: Palette 39/53

JEE Advanced 1996 MCQ

The angle between a pair of tangents drawn from a point P to the circle ${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\, + \,9\,{\sin ^2}\,\alpha \, + \,13\,{\cos ^2}\,\alpha \, = \,0$ is $2\,\alpha $.
The equation of the locus of the point P is

A

${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\, + \,4\, = \,0$

B

${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\,\, - \,9\,\, = \,0$

C

${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\,\, - \,4\,\, = \,0$

D

${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\,\, + \,9\,\, = \,0$

Question 40: Palette 40/53

JEE Advanced 1994 MCQ

The circles ${x^2} + {y^2} - 10x + 16 = 0$ and ${x^2} + {y^2} = {r^2}$ intersect each other in two distinct points if

A

r < 2

B

r > 8

C

2 < r < 8

D

$2 \le r \le 8$

Question 41: Palette 41/53

JEE Advanced 1993 MCQ

The locus of the centre of a circle, which touches externally the circle ${x^2} + {y^2} - 6x - 6y + 14 = 0$ and also touches the y-axis, is given by the equation:

A

${x^2} - 6x - 10y + 14 = 0$

B

${x^2} - 10x - 6y + 14 = 0$

C

${y^2} - 6x - 10y + 14 = 0$

D

${y^2} - 10x - 6y + 14 = 0$

Question 42: Palette 42/53

JEE Advanced 1992 MCQ

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

A

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

B

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

C

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

D

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

Question 43: Palette 43/53

JEE Advanced 1989 MCQ

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

A

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

B

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

C

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

D

${x^2} + {y^2} - 2x + 2y = 62$c

Question 44: Palette 44/53

JEE Advanced 1989 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 points, then

A

2 < r < 8

B

r < 2

C

r = 2

D

r > 2

Question 45: Palette 45/53

JEE Advanced 1989 MCQ

The line x + 3y = 0 is a diameter of the circle ${x^2} + {y^2} - 6x + 2y = 0\,$.

A

TRUE

B

FALSE

Question 46: Palette 46/53

JEE Advanced 1988 MCQ

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

A

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

B

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

C

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

D

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

Question 47: Palette 47/53

JEE Advanced 1988 MSQ

The equations of the tangents drawn from the origin to the circle ${x^2}\, + \,{y^2}\, - \,2rx\,\, - 2hy\, + {h^2} = 0$, are

A

x = 0

B

y = 0

C

$({h^2}\, - \,{r^2})\,x - \,\,2rhy\, = \,0$

D

$({h^2}\, - \,{r^2})\,x + \,\,2rhy\, = \,0$

Question 48: Palette 48/53

JEE Advanced 1985 MCQ

No tangent can be drawn from the point (5/2, 1) to the circumcircle of the triangle with vertices $\left( {1,\sqrt 3 } \right)\,\,\left( {1, - \sqrt 3 } \right),\,\,\left( {3,\sqrt 3 } \right)$.

A

TRUE

B

FALSE

Question 49: Palette 49/53

JEE Advanced 1984 MCQ

The locus of the mid-point of a chord of the circle ${x^2} + {y^2} = 4$ which subtends a right angle at the origin is

A

x + y = 2

B

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

C

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

D

$x + y $=1

Question 50: Palette 50/53

JEE Advanced 1983 MCQ

The equation of the circle passing through (1, 1) and the points of intersection of ${x^2} + {y^2} + 13x - 3y = 0$ and $2{x^2} + 2{y^2} + 4x - 7y - 25 = 0$ is

A

$4{x^2} + 4{y^2} - 30x - 10y - 25 = 0$

B

$4{x^2} + 4{y^2} + 30x - 13y - 25 = 0$

C

$4{x^2} + 4{y^2} - 17x - 10y + 25 = 0$

D

none of these

Question 51: Palette 51/53

JEE Advanced 1983 MCQ

The centre of the circle passing through the point (0, 1) and touching the curve $\,y = {x^2}$ at (2, 4) is

A

$\left( {{{ - 16} \over 5},{{ - 27} \over {10}}} \right)$

B

$\left( {{{ - 16} \over 7},{{53} \over {10}}} \right)$

C

$\left( {{{ - 16} \over 5},{{53} \over {10}}} \right)$

D

none of these

Question 52: Palette 52/53

JEE Advanced 1980 MCQ

Two circles ${x^2} + {y^2} = 6$ and ${x^2} + {y^2} - 6x + 8 = 0$ are given. Then the equation of the circle through their points of intersection and the point (1, 1) is

A

${x^2} + {y^2} - 6x + 4 = 0$

B

${x^2} + {y^2} - 3x + 1 = 0$

C

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

D

none of these

Question 53: Palette 53/53

JEE Advanced 1980 MCQ

A square is inscribed in the circle ${x^2} + {y^2} - 2x + 4y + 3 = 0$. Its sides are parallel to the coordinate axes. The one vertex of the square is

A

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

B

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

C

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

D

none of these