# JEE Main > Circle and System of Circles

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

Physics
Chemistry
Maths

Q 1.

Correct4

Incorrect-1

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

A

{tex} 0 < k < 1 / 2 {/tex}

{tex} k \geq 1 / 2 {/tex}

C

{tex} - 1 / 2 \leq k \leq 1 / 2 {/tex}

D

{tex} k \leq 1 / 2 {/tex}

##### Explanation

Q 2.

Correct4

Incorrect-1

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

A

{tex} ( 3 , - 4 ) {/tex}

B

{tex} ( - 3,4 ) {/tex}

{tex} ( - 3 , - 4 ) {/tex}

D

{tex} ( 3,4 ) {/tex}

##### Explanation

Q 3.

Correct4

Incorrect-1

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

A

all values of {tex} p {/tex}

B

all except one value of {tex} p {/tex}

all except two values of {tex} p {/tex}

D

exactly one value of {tex} p {/tex}

##### Explanation

Q 4.

Correct4

Incorrect-1

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

{tex} - 35 < m < 15 {/tex}

B

{tex} 15 < m < 65 {/tex}

C

{tex} 35 < m < 85 {/tex}

D

{tex} - 85 < m < - 35 {/tex}

##### Explanation

Q 5.

Correct4

Incorrect-1

The two circles {tex} x ^ { 2 } + y ^ { 2 } = a x {/tex} and {tex} x ^ { 2 } + y ^ { 2 } = c ^ { 2 } ( c > 0 ) {/tex} touch each other if

{tex} | a | = c {/tex}

B

{tex} a = 2 c {/tex}

C

{tex} | a | = 2 c {/tex}

D

{tex} 2 | a | = c {/tex}

##### Explanation

Q 6.

Correct4

Incorrect-1

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

{tex} \frac { 10 } { 3 } {/tex}

B

{tex} \frac { 3 } { 5 } {/tex}

C

{tex} \frac { 6 } { 5 } {/tex}

D

{tex} \frac { 5 } { 3 } {/tex}

##### Explanation

Q 7.

Correct4

Incorrect-1

The circle passing through {tex} ( 1 , - 2 ) {/tex} and touching the axis of {tex} x {/tex} at {tex} ( 3,0 ) {/tex} also passes through the point

A

{tex} ( 2 , - 5 ) {/tex}

{tex} ( 5 , - 2 ) {/tex}

C

{tex} ( - 2,5 ) {/tex}

D

{tex} ( - 5,2 ) {/tex}

##### Explanation

Q 8.

Correct4

Incorrect-1

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

A

{tex} \frac { 1 } { 2 } {/tex}

{tex} \frac { 1 } { 4 } {/tex}

C

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

D

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

##### Explanation

Q 9.

Correct4

Incorrect-1

If the point {tex} ( 1,4 ) {/tex} lies inside the circle {tex} x ^ { 2 } + y ^ { 2 } - 6 x - 10 y + p = 0 {/tex} and the circle does not touch or intersect the coordinate axes, then the set of all possible values of {tex} p {/tex} is the interval:

A

{tex} ( 0,25 ) {/tex}

B

{tex} ( 25,39 ) {/tex}

C

{tex} ( 9,25 ) {/tex}

{tex} ( 25,29 ) {/tex}

##### Explanation

Q 10.

Correct4

Incorrect-1

The set of all real values of {tex} \lambda {/tex} for which exactly two common tangents can be drawn to the circles {tex} x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 6 = 0 {/tex} and {tex} x ^ { 2 } + y ^ { 2 } - 10 x - 10 y + \lambda = 0 {/tex} is the interval

A

{tex} ( 12,32 ) {/tex}

{tex} ( 18,42 ) {/tex}

C

{tex} ( 12,24 ) {/tex}

D

{tex} ( 18,48 ) {/tex}

##### Explanation

Q 11.

Correct4

Incorrect-1

For the two circles {tex} x ^ { 2 } + y ^ { 2 } = 16 {/tex} and {tex} x ^ { 2 } + y ^ { 2 } - 2 y = 0 {/tex} , there is/are

A

one pair of common tangents

B

two pairs of common tangents

C

three common tangents

no common tangent

##### Explanation

Q 12.

Correct4

Incorrect-1

The equation of the circle described on the chord {tex} 3 x + y + 5 = 0 {/tex} of the circle {tex} x ^ { 2 } + y ^ { 2 } = 16 {/tex} as diameter is

{tex} x ^ { 2 } + y ^ { 2 } + 3 x + y - 11 = 0 {/tex}

B

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

C

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

D

{tex} x ^ { 2 } + y ^ { 2 } + 3 x + y - 22 = 0 {/tex}

##### Explanation

Q 13.

Correct4

Incorrect-1

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

A

2

3

C

4

D

1

##### Explanation

Q 14.

Correct4

Incorrect-1

If {tex} y + 3 x = 0 {/tex} is the equation of a chord of the circle {tex} x ^ { 2 } + y ^ { 2 } - 30 x = 0 {/tex} , then the equation of the circle with this chord as diameter is:

A

{tex} x ^ { 2 } + y ^ { 2 } + 3 x + 9 y = 0 {/tex}

{tex} x ^ { 2 } + y ^ { 2 } - 3 x + 9 y = 0 {/tex}

C

{tex} x ^ { 2 } + y ^ { 2 } - 3 x - 9 y = 0 {/tex}

D

{tex} x ^ { 2 } + y ^ { 2 } + 3 x - 9 y = 0 {/tex}

##### Explanation

Q 15.

Correct4

Incorrect-1

If the incentre of an equilateral triangle is {tex} ( 1,1 ) {/tex} and the equation of its one side is {tex} 3 x + 4 y + 3 = 0 {/tex}, then the equation of the circumcircle of this triangle is

A

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

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

C

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

D

{tex} x ^ { 2 } + y ^ { 2 } - 2 x - 2 y - 7 = 0 {/tex}

##### Explanation

Q 16.

Correct4

Incorrect-1

If a circle passing through the point {tex} ( - 1,0 ) {/tex} touches {tex} y {/tex} -axis at {tex} ( 0,2 ) , {/tex} then the length of the chord of the circle along the {tex} x {/tex} -axis is

A

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

B

{tex} \frac { 5 } { 2 } {/tex}

3

D

5

##### Explanation

Q 17.

Correct4

Incorrect-1

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

A

10

B

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

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

D

5

##### Explanation

Q 18.

Correct4

Incorrect-1

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

a parabola

B

a circle

C

an ellipse which is not a circle

D

a hyperbola

##### Explanation

Q 19.

Correct4

Incorrect-1

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

{tex} 2 x - 3 y + 10 = 0 {/tex}

B

{tex} 3 x + 4 y - 3 = 0 {/tex}

C

{tex} 4 x + 5 y - 6 = 0 {/tex}

D

{tex} 5 x + 2 y + 4 = 0 {/tex}

##### Explanation

Q 20.

Correct4

Incorrect-1

Equation of the tangent to the circle, at the point {tex} ( 1 , - 1 ) {/tex}, whose centre is the point of intersection of the straight lines {tex} x - y = 1 {/tex} and {tex} 2 x + y = 3 {/tex} is

{tex} x + 4 y + 3 = 0 {/tex}

B

{tex} 3 x - y - 4 = 0 {/tex}

C

{tex} x - 3 y - 4 = 0 {/tex}

D

{tex} 4 x + y - 3 = 0 {/tex}

##### Explanation

Q 21.

Correct4

Incorrect-1

If the straight line {tex} m x - y = 1 + 2 x {/tex} intersects the circle {tex} x ^ { 2 } + y ^ { 2 } = 1 {/tex} at least at one point, then the set of values of {tex} m {/tex} is

A

{tex} \left[ - \frac { 4 } { 3 } , 0 \right] {/tex}

B

{tex} \left[ - \frac { 4 } { 3 } , \frac { 4 } { 3 } \right] {/tex}

C

{tex} \left[ 0 , \frac { 4 } { 3 } \right] {/tex}

All of these

##### Explanation

Q 22.

Correct4

Incorrect-1

Circles are drawn having the sides of triangle {tex} A B C {/tex} as their diameters. Radical centre of the circles is the

A

circumcentre of triangle {tex} A B C {/tex}

B

in-centre of triangle {tex} A B C {/tex}

orthocentre of triangle {tex} A B C {/tex}

D

centroid of triangle {tex} A B C {/tex}

##### Explanation

Q 23.

Correct4

Incorrect-1

The circle described on the line joining the points {tex} ( 0,1 ) , ( a , b ) {/tex} as a diameter cuts the {tex} x {/tex} -axis at the points whose abscissa are roots of the equation

A

{tex} x ^ { 2 } + a x + b = 0 {/tex}

{tex} x ^ { 2 } - a x + b = 0 {/tex}

C

{tex} x ^ { 2 } + a x - b = 0 {/tex}

D

{tex} x ^ { 2 } - a x - b = 0 {/tex}

##### Explanation

Q 24.

Correct4

Incorrect-1

The straight line {tex} y = m x + c \ {/tex} cuts the circle {tex} x ^ { 2 } + y ^ { 2 } = a ^ { 2 } {/tex} at the real points if

A

{tex} \sqrt { a ^ { 2 } \left( 1 + m ^ { 2 } \right) } \leq | c | {/tex}

B

{tex} \sqrt { a ^ { 2 } \left( 1 - m ^ { 2 } \right) } \leq | c | {/tex}

{tex} \sqrt { a ^ { 2 } \left( 1 + m ^ { 2 } \right) } \geq | c | {/tex}

D

{tex} \sqrt { a ^ { 2 } \left( 1 - m ^ { 2 } \right) } \geq | c | {/tex}

##### Explanation

Q 25.

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 } , - \sqrt { 2 } \right) {/tex}