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JEE Advanced > Circle and System of Circles

Explore popular questions from Circle and System of Circles for JEE Advanced. This collection covers Circle and System of Circles previous year JEE Advanced questions hand picked by experienced teachers.

Q 1.

Correct4

Incorrect-1

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

A

{tex} ( 1 + \sqrt { 2 } , - 2 ) {/tex}

B

{tex} ( 1 - \sqrt { 2 } , - 2 ) {/tex}

C

{tex} ( 1 , - 2 + \sqrt { 2 } ) {/tex}

none of these

Explanation


Q 2.

Correct4

Incorrect-1

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

A

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

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

C

{tex} x ^ { 2 } + y ^ { 2 } - 4 y + 2 = 0 {/tex}

D

none of these

Explanation

Q 3.

Correct4

Incorrect-1

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

A

{tex} \left( \frac { - 16 } { 5 } , \frac { 27 } { 10 } \right) {/tex}

B

{tex} \left( \frac { - 16 } { 7 } , \frac { 53 } { 10 } \right) {/tex}

{tex} \left( \frac { - 16 } { 5 } , \frac { 53 } { 10 } \right) {/tex}

D

none of these

Explanation


Q 4.

Correct4

Incorrect-1

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

A

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

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

C

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

D

none of these

Explanation


Q 5.

Correct4

Incorrect-1

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

A

{tex} x + y = 2 {/tex}

B

{tex} x ^ { 2 } + y ^ { 2 } = 1 {/tex}

{tex} x ^ { 2 } + y ^ { 2 } = 2 {/tex}

D

{tex} x + y = 1 {/tex}

Explanation

Q 6.

Correct4

Incorrect-1

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

{tex} 2 a x + 2 b y - \left( a ^ { 2 } + b ^ { 2 } + k ^ { 2 } \right) = 0 {/tex}

B

{tex} 2 a x + 2 b y - \left( a ^ { 2 } - b ^ { 2 } + k ^ { 2 } \right) = 0 {/tex}

C

{tex} x ^ { 2 } + y ^ { 2 } - 3 a x - 4 b y + \left( a ^ { 2 } + b ^ { 2 } - k ^ { 2 } \right) = 0 {/tex}

D

{tex} x ^ { 2 } + y ^ { 2 } - 2 a x - 3 b y + \left( a ^ { 2 } - b ^ { 2 } - k ^ { 2 } \right) = 0 {/tex}

Explanation



Q 7.

Correct4

Incorrect-1

If the two circles {tex} ( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = r ^ { 2 } {/tex} and {tex} x ^ { 2 } + y ^ { 2 } - 8 x + 2 y + 8 = 0 {/tex} intersect in two distinct points, then

{tex} 2 < r < 8 {/tex}

B

{tex} r < 2 {/tex}

C

{tex} r = 2 {/tex}

D

{tex} r > 2 {/tex}

Explanation


Q 8.

Correct4

Incorrect-1

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

A

{tex} x ^ { 2 } + y ^ { 2 } + 2 x - 2 y = 62 {/tex}

B

{tex} x ^ { 2 } + y ^ { 2 } + 2 x - 2 y = 47 {/tex}

{tex} x ^ { 2 } + y ^ { 2 } - 2 x + 2 y = 47 {/tex}

D

{tex} x ^ { 2 } + y ^ { 2 } - 2 x + 2 y = 62 {/tex}

Explanation

Q 9.

Correct4

Incorrect-1

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

A

{tex} \left( \frac { 3 } { 2 } , \frac { 1 } { 2 } \right) {/tex}

B

{tex} \left( \frac { 1 } { 2 } , \frac { 3 } { 2 } \right) {/tex}

C

{tex} \left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right) {/tex}

{tex} \left( \frac { 1 } { 2 } , - 2 ^ { \frac { 1 } { 2 } } \right) {/tex}

Explanation

Q 10.

Correct4

Incorrect-1

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

A

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

B

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

C

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

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

Explanation

Q 11.

Correct4

Incorrect-1

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

A

{tex} r < 2 {/tex}

B

{tex} r > 8 {/tex}

{tex} 2 < r < 8 {/tex}

D

{tex} 2 \leq r \leq 8 {/tex}

Explanation

Q 12.

Correct4

Incorrect-1

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

A

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

B

{tex} x ^ { 2 } + y ^ { 2 } + 4 x - 6 y - 9 = 0 {/tex}

C

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

{tex} x ^ { 2 } + y ^ { 2 } + 4 x - 6 y + 9 = 0 {/tex}

Explanation


Q 13.

Correct4

Incorrect-1

If two distinct chords, drawn from the point {tex} ( p , q ) {/tex} on the circle {tex} x ^ { 2 } + y ^ { 2 } = p x + q y ( \text { where } p q \neq 0 ) {/tex} are bisected by the {tex} x {/tex} -axis, then

A

{tex} p ^ { 2 } = q ^ { 2 } {/tex}

B

{tex} p ^ { 2 } = 8 q ^ { 2 } {/tex}

C

{tex} P ^ { 2 } < 8 q ^ { 2 } {/tex}

{tex} p ^ { 2 } > 8 q ^ { 2 } {/tex}

Explanation



Q 14.

Correct4

Incorrect-1

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

A

{tex} \frac { \pi } { 2 } {/tex}

B

{tex} \frac { \pi } { 3 } {/tex}

{tex} \frac { \pi } { 4 } {/tex}

D

{tex} \frac { \pi } { 6 } {/tex}

Explanation


Q 15.

Correct4

Incorrect-1

If the circles {tex} x ^ { 2 } + y ^ { 2 } + 2 x + 2 k y + 6 = 0 , x ^ { 2 } + y ^ { 2 } + 2 k y + k = 0 {/tex} intersect orthogonally, then {tex} k {/tex} is

{tex} 2 {/tex} or {tex} - \frac { 3 } { 2 } {/tex}

B

{tex} - 2 {/tex} or {tex} - \frac { 3 } { 2 } {/tex}

C

{tex}2{/tex} or {tex} \frac { 3 } { 2 } {/tex}

D

{tex} - 2 {/tex} or {tex} \frac { 3 } { 2 } {/tex}

Explanation

Q 16.

Correct4

Incorrect-1

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

A

a parabola

a circle

C

an ellipse

D

a pair of straight lines

Explanation


Q 17.

Correct4

Incorrect-1

Let {tex} P Q {/tex} and {tex} R S {/tex} be tangents at the extremities of the diameter {tex} P R {/tex} of a circle of radius {tex} r {/tex}. If {tex} P S {/tex} and {tex} R Q {/tex} intersect at a point {tex} X {/tex} on the circumference of the circle, then {tex} 2 r {/tex} equals

{tex} \sqrt { P Q \cdot R S } {/tex}

B

{tex} ( P Q + R S ) / 2 {/tex}

C

{tex} 2 P Q \cdot R S / ( P Q + R S ) {/tex}

D

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

Explanation


Q 18.

Correct4

Incorrect-1

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

A

{tex} 4 {/tex}

B

{tex} 2 \sqrt { 5 } {/tex}

{tex} 5 {/tex}

D

{tex} 3 \sqrt { 5 } {/tex}

Explanation