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JEE • MATHEMATICS • SYSTEM OF CIRCLES

System of circles practice · Medium

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Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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1 Medium

1 of 19

Application +4 | -1

If a circle passes through the point (1, 2) and cuts the circle

{x^2} + {y^2} = 4

orthogonally, then the equation of the locus of its centre is

View answer and explanation

Correct option: C

2x + 4y - 9 = 0

Option C (

2x + 4y - 9 = 0

) is correct because it represents the locus of the center of a circle passing through (1, 2) and intersecting the given circle orthogonally.

2 Medium

2 of 19

Application +4 | -1

The equation of the circle which passes through the origin and cuts orthogonally each of the circles

{x^2} + {y^2} - 6x + 8 = 0

and

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

View answer and explanation

Correct option: B

3{x^2} + 3{y^2} - 8x + 29y = 0

Option C (

(7, - 5)

) is correct because it represents the other end of the diameter, calculated using the midpoint formula with the given point and the center of the circle.

3 Medium

3 of 19

Application +4 | -1

If a chord of a circle

x^2+y^2=25

with one extremity at

(4,3)

subtends a right angle at the centre of this circle, then the coordinates of the other extremity of this chord can be.

View answer and explanation

Correct option: D

(-3,4)

Option D ((-3,4)) is correct because the dot product of the position vectors of the two extremities of the chord is zero, indicating a right angle at the center.

4 Medium

4 of 19

Analyzing +4 | -1

The circle

{x^2} + {y^2} = 4

cuts the circle

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

at the points

A

and

B

. If the circle

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

passes through

A

and

B

then the value of

k

View answer and explanation

Correct option: D

Option D (4) is correct because by substituting the intersection points of the first two circles into the third circle's equation, we find that k equals 4.

5 Medium

5 of 19

Analyzing +4 | -1

If the circles

{x^2} + {y^2} + 5Kx + 2y + K = 0

and

2\left( {{x^2} + {y^2}} \right) + 2Kx + 3y - 1 = 0,\left( {K \in R} \right),

intersect at the points

P

and

Q,

then the line

4x + 5y - K = 0

passes through

P

and

Q,

for

View answer and explanation

Correct option: A

No value of K

Option A (No value of K) is correct because the condition for the line to pass through the intersection points of the circles leads to conflicting values of K.

6 Medium

6 of 19

Application +4 | -1

If the circle

{x^2} + {y^2} + 4x + 22y + c = 0\;\;

bisects the circumference of the circle

{x^2} + {y^2} - 2x + 8y - d = 0

, then

c + d

is equal to

View answer and explanation

Correct option: B

Option B (50) is correct because the sum c+d is found by using the fact that the common chord passes through the center of the second circle.

7 Medium

7 of 19

Application +4 | -1

The limiting points of the coaxial system of circles

{x^2} + {y^2} + 2\lambda y + 4 = 0

are

View answer and explanation

Correct option: D

\left( {0, \pm 2} \right)

Option D (

(0, \pm 2)

) is correct because it represents the limiting points of the coaxial system of circles, found by setting the radius to zero.

8 Medium

8 of 19

Application +4 | -1

The equation of the circle on the common chord of the circles

{\left( {x - a} \right)^2} + {y^2} = {a^2}

and

{x^2} + {\left( {y + b} \right)^2} = {b^2}

as diameter, is

View answer and explanation

Correct option: C

\left( {{a^2} + {b^2}} \right)\left( {{x^2} + {y^2}} \right) = 2ab\left( {bx - ay} \right)

Option C (

\left( {{a^2} + {b^2}} \right)\left( {{x^2} + {y^2}} \right) = 2ab\left( {bx - ay} \right)

) is correct because it represents the equation of the circle on the common chord as diameter.

9 Medium

9 of 19

Application +4 | -1

Equation of the circle which passes through the origin, has its centre on the line

x+y=4

and cuts the circle

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

orthogonally, is

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it satisfies the given conditions: passing through the origin, having its center on the line

x+y=4

, and cutting the given circle orthogonally.

10 Medium

10 of 19

Application +4 | -1

Line

L

touches circles

C_1, C

and

C_2

having radii

r_1, r, r_2

respectively. These circle pairwise touch each-other externally. If

r_1=36

and

r_2=900

then find the value of

\sqrt{r}

View answer and explanation

Correct option: A

Option A (5) is correct because the radius of the middle circle is 25, and the square root of 25 is 5.

11 Medium

11 of 19

Analyzing +4 | -1

\mathrm{P}

and

\mathrm{Q}

are the points of intersection of the circles

x^{2}+y^{2}+3 x+7 y+2 p-5=0

and

x^{2}+y^{2}+2 x+2 y-p^{2}=0

then there is a circle passing through

\mathrm{P}, \mathrm{Q}

and

(1,1)

for:

View answer and explanation

Correct option: A

all except one value of

p

Option A (all except one value of

p

) is correct because the equation for

\lambda

is undefined for

p = -1

12 Medium

12 of 19

Analyzing +4 | -1

The number of circles belonging to the system of circles

2\left( {{x^2} + {y^2}} \right) + \lambda \;x - \left( {1 + {\lambda ^2}} \right)y - 10 = 0

and orthogonal to

{x^2} + {y^2} + 4x + 6y + 3 = 0,

View answer and explanation

Correct option: A

The correct answer is A (2). By finding the condition for orthogonality between the circles, a quadratic equation in

\lambda

is obtained, yielding two possible values for

\lambda

, hence two circles.

13 Medium

13 of 19

Analyzing +4 | -1

The condition for the coaxial system

{x^2} + {y^2} + 2\lambda x + c = 0

, where

\lambda

is a parameter and

c

is a constant to have distinct limiting points, is

View answer and explanation

Correct option: D

c > 0

The correct answer is

c > 0

because the condition for distinct limiting points in a coaxial system requires the radius of the circles to be real, which implies that

c

must be positive.

14 Medium

14 of 19

Application +4 | -1

If the circle

{x^2} + {y^2} + 6x - 2y + k = 0

bisects the circumference of the circle

{x^2} + {y^2} + 2x - 6y - 15 = 0

, then

k

is equal to

View answer and explanation

Correct option: D

- 23

Option D (

- 23

) is correct because it is derived from the condition that the first circle bisects the circumference of the second circle, implying that the common chord passes through the center of the second circle.

15 Medium

15 of 19

Application +4 | -1

The locus of the centres of the circles which cut the circles

x^2+y^2+4 x-6 y+9=0

and

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

orthogonally is

View answer and explanation

Correct option: C

9 x-10 y+11=0

Option C (

9 {x}-10 {y}+11=0

) is correct because it is the equation of the line obtained by subtracting the equations of the two circles after applying the condition for orthogonality.

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16 Medium

16 of 19

Application +4 | -1

Let

AB

be the intercept of the line

y = x

the circle

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

. Then, the equation of the circle with

AB

as its diameter is

View answer and explanation

Correct option: A

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

The correct answer is A (

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

) because it represents the equation of a circle with the intercept AB as its diameter, where A and B are the intersection points of the line and the given circle.

17 Medium

17 of 19

Application +4 | -1

The limiting points of coaxial-system determined by the circles

{x^2} + {y^2} + 5x + y + 4 = 0

and

{x^2} + {y^2} + 10x - 4y - 1 = 0

are

View answer and explanation

Correct option: B

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

and

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

Option B (

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

and

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

) is correct because these are the limiting points found by setting the radius of the coaxial system of circles to zero.

18 Medium

18 of 19

Application +4 | -1

The mid point of chord by the circle

x^2+y^2=16

on the line

x+y+1=0

View answer and explanation

Correct option: B

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

The correct answer is (-1/2, -1/2) because it is the midpoint of the chord formed by the intersection of the circle and the line.

19 Medium

19 of 19

Analyzing +4 | -1

If common chord of the circle

{C}

with centre at

(2,1)

and radius

{r}

and the circle

x^2+y^2-2 x-6 y+6=0

is a diameter of the second circle, then value of

{r}

View answer and explanation

Correct option: A

Option A (3) is correct because by finding the equation of the common chord and using the fact that it passes through the center of the second circle, we can solve for the radius r.

Revision notes and key takeaways

Review SYSTEM OF CIRCLES in your textbook or class notes, then use this set to check your understanding.

Key Takeaways

✓ This set contains 19 single-correct-answer MCQs on SYSTEM OF CIRCLES.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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