If the circle ${x^2} + {y^2} + 2x + 3y + 1 = 0$ cuts ${x^2} + {y^2} + 4x + 3y + 2 = 0$ in $A$ and $B$, then the equation of the circle on $AB$ as diameter is

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

${x^2} + {y^2} + x + 6y + 1 = 0$

Back to SYSTEM OF CIRCLES

Your Progress

0 / 98 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • SYSTEM OF CIRCLES

System of circles practice · Mixed difficulty

Try a question, then open “View answer” to check it. Sign in to submit answers and save progress.

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.

Take a chapter quiz →

Showing 31-60 of 98 questions

31 Easy

31 of 98

Application +4 | -1

The equation of the common tangent of the two touching circles,

{y^2} + {x^2} - 6x - 12y + 37 = 0

and

{x^2} + {y^2} - 6y + 7 = 0

View answer and explanation

Correct option: C

x - y - 5 = 0

Option C (

x - y - 5 = 0

) is correct because it is the equation obtained by subtracting the equations of the two circles.

32 Easy

32 of 98

Analyzing +4 | -1

P

and

Q

are the points of intersection of the circles

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

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

, then there is a circle passing through

P,Q

and

\left( {1,1} \right)

and

View answer and explanation

Correct option: B

All except one value of

p

Option B (All except one value of

p

) is correct because there is a circle passing through P, Q, and (1,1) for all values of p except when the equation becomes undefined.

33 Easy

33 of 98

Application +4 | -1

S \equiv x^2+y^2+2 x+3 y+1=0

and

S^{\prime} \equiv x^2+y^2+4 x+3 y+2=0

are two circles, then the point

(-3,-2)

lies

View answer and explanation

Correct option: A

inside

S^{\prime}

only

Option A (inside

S^{\prime}

only) is correct because substituting the point into the circle equations shows it's outside S and inside S'.

34 Easy

34 of 98

Application +4 | -1

The point from which the tangents to the circles

x^2+y^2-8 x+40=0

5 x^2+5 y^2-25 x+80=0

x^2+y^2-8 x+16 y+160=0

are equal in length, is

View answer and explanation

Correct option: C

(8,-15 / 2)

Option C (

(8,-15 / 2)

) is correct because it represents the radical center of the three circles, where the lengths of tangents from that point to each circle are equal.

35 Easy

35 of 98

Application +4 | -1

The equation of the circle passing through (1, -3) and the points common to the two circles

x^2+y^2-6 x+8 y-16=0, x^2+y^2+4 x-2 y-8=0

, is

View answer and explanation

Correct option: B

2 x^2+2 y^2+3 x+y-20=0

Option B (

2 x^2+2 y^2+3 x+y-20=0

) is correct because it represents the equation of a circle passing through the given point and the intersection of the two given circles.

36 Medium

36 of 98

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.

37 Easy

37 of 98

Analyzing +4 | -1

The locus of the centres of circles passing through the origin and intersecting the fixed circle

x^2+y^2-5 x+3 y-1=0

orthogonally is

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the locus of the centers is a straight line with the equation 5x - 3y = -1, which does not match any of the other options.

38 Easy

38 of 98

Application +4 | -1

If the circle

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

cuts

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

A

and

B

, then the equation of the circle on

AB

as diameter is

View answer and explanation

Correct option: B

2{x^2} + 2{y^2} + 2x + 6y + 1 = 0

Option D (

2{x^2} + 2{y^2} + 2x + 6y + 1 = 0

) is correct because it represents the equation of the circle with the common chord as a diameter, found by using the equation of the common chord and the equation of the first circle.

39 Easy

39 of 98

Application +4 | -1

The equation of a circle which passes through

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

and whose radical axis in relation to the circle

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

x = a/2,

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it represents the equation of a circle passing through (2a,0) and having the given radical axis.

40 Easy

40 of 98

Application +4 | -1

The radical axis of the circles, belonging to the coaxal system of circles whose limiting points are

(1,3)

and

(2,6)

, is

View answer and explanation

Correct option: B

x+3 y-15=0

Option B (

x+3 y-15=0

) is correct because it represents the radical axis of the coaxial system of circles with the given limiting points.

41 Easy

41 of 98

Application +4 | -1

The two circles

{x^2} + {y^2} - 2x + 22y + 5 = 0

and

{x^2} + {y^2} + 14x + 6y + k = 0

intersect orthogonally provided

k

is equal to

View answer and explanation

Correct option: A

Option A (47) is correct because when two circles intersect orthogonally, the condition

2g_1g_2 + 2f_1f_2 = c_1 + c_2

must be satisfied, and solving for k yields 47.

42 Easy

42 of 98

Application +4 | -1

The limiting points of the system of circles represented by the equation

2\left(x^2+y^2\right)+\lambda x+\frac{9}{2}=0

, are

View answer and explanation

Correct option: A

\left(\pm \frac{3}{2}, 0\right)

Option A (

\left(\pm \frac{{3}}{{2}}, 0\right)

) is correct because it represents the limiting points of the given system of circles, which are the centers of the point circles.

43 Easy

43 of 98

Application +4 | -1

If the point

(3,4)

lies inside and the point

(-3,-4)

lies outside the circle

x^2+y^2-7 x+5 y-p=0

, then the set of all possible values of

p

View answer and explanation

Correct option: C

(24,26)

The correct answer is (24,26) because the point (3,4) lies inside the circle and (-3,-4) lies outside the circle, which gives the range of p.

44 Medium

44 of 98

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.

45 Easy

45 of 98

Application +4 | -1

The length of the common chord of two circles of radii 15 and 20, whose centres are 25 units apart, is

View answer and explanation

Correct option: A

Option A (24) is correct because the circles intersect orthogonally, and the length of the common chord can be calculated using the formula

\frac{2rR}{\sqrt{r^2+R^2}}

Explore Other Chapters in MATHEMATICS

View All →

SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

46 Medium

46 of 98

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.

47 Easy

47 of 98

Application +4 | -1

(1,2)

is a limiting point of a coaxial system of circles containing the circle

x^2+y^2+x-5 y+9=0

, then the equation of the radical axis, is

View answer and explanation

Correct option: B

3 x-y+4=0

Option B (

3 x-y+4=0

) is correct because it represents the equation of the radical axis of the coaxial system of circles, given one limiting point and one circle.

48 Easy

48 of 98

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 when one circle bisects the circumference of another, the common chord passes through the center of the second circle, leading to the equation

c + d = 50

49 Easy

49 of 98

Application +4 | -1

If the circles

{x^2} + {y^2} + 2x + 2ky + 6 = 0

and

{x^2} + {y^2} + 2ky + K = 0

intersect orthogonally, then

k

View answer and explanation

Correct option: A

2\,\,\mathrm{or} - \frac{3}{2}

Option A (

2\,\,\mathrm{or} - \frac{3}{2}

) is correct because these are the values of k that satisfy the orthogonality condition for the two circles.

50 Easy

50 of 98

Analyzing +4 | -1

Circle

{x^2} + {y^2} - 2x - \lambda x - 1 = 0

passes through two fixed points coordinates of the points are

View answer and explanation

Correct option: A

\left( {0,\; \pm 1} \right)

Option A

\left( {0,\; \pm 1} \right)

is correct because the circle passes through the intersection points of the line

x=0

and the given circle equation.

51 Easy

51 of 98

Application +4 | -1

The equation of the circle, which cuts orthogonally each of three circles

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

{x^2} + {y^2} + 5x - 5y + 9 = 0

and

{x^2} + {y^2} + 7x - 9y + 29 = 0

View answer and explanation

Correct option: A

{x^2} + {y^2} - 16x - 18y - 4 = 0

Option A (

{x^2} + {y^2} - 16x - 18y - 4 = 0

) is correct because it represents the equation of a circle that cuts the three given circles orthogonally.

52 Easy

52 of 98

Analyzing +4 | -1

The coordinates of the centre of the circle which intersects circles

{x^2} + {y^2} + 4x + 7 = 0,\;2{x^2} + 2{y^2} + 3x + 5y + 9 = 0

and

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

orthogonally are

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the center of the circle intersecting the given circles orthogonally is the radical center of the given circles.

53 Easy

53 of 98

Application +4 | -1

The equation of the circle passing through the point

\left( {1,1} \right)

and through the points of intersection of the circles

{x^2} + {y^2} = 6

amnd

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

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it represents the equation of a circle passing through (1, 1) and the intersection points of the given circles.

54 Medium

54 of 98

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.

55 Easy

55 of 98

Application +4 | -1

An equation of the circle through

(1,1)

and the points of intersection of

x^2+y^2+13 x-3 y=0

and

2 x^2+2 y^2+4 x-7 y-25=0

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it represents a circle passing through (1,1) and the intersection points of the given circles.

56 Easy

56 of 98

Application +4 | -1

The equation of the circle and its chord are respectively

x^2+y^2=a^2

and

x \cos \alpha+y \sin \alpha=p

. The equation of the circle of which this chord is a diameter, is

View answer and explanation

Correct option: A

x^2+y^2-2 p x \cos \alpha-2 p y \sin \alpha+2 p^2-a^2=0

Option A (

x^2+y^2-2 p x \cos \alpha-2 p y \sin \alpha+2 p^2-a^2=0

) is correct because it represents the equation of a circle where the given chord is a diameter, derived using the concept of a family of circles.

57 Medium

57 of 98

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

58 Medium

58 of 98

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.

59 Easy

59 of 98

Analyzing +4 | -1

The circle

{S_1}

with centre

{C_1}\left( {{a_1},{b_1}} \right)

and radius

{r_1}

touches externally the circle

{S_2}

with centre

{C_2}\left( {{a_2},{b_2}} \right)

and radius

{r_2}

. If the tangent at their common point passes through the origin, then

View answer and explanation

Correct option: B

\left( {a_1^2 - a_2^2} \right) + \left( {b_1^2 - b_2^2} \right) = r_1^2 - r_2^2

Option B (

\left( {a_1^2 - a_2^2} \right) + \left( {b_1^2 - b_2^2} \right) = r_1^2 - r_2^2

) is correct because it is derived from the condition that the common tangent passes through the origin.

60 Easy

60 of 98

Application +4 | -1

The locus of centres of family of circle passing through the origin and cutting the circle

{x^2} + {y^2} + 4x - 6y - 13 = 0

orthogonally, is

View answer and explanation

Correct option: D

4x - 6y - 13 = 0

Option D is correct because the locus of the centers is found by using the orthogonality condition of circles, resulting in the equation 4x - 6y - 13 = 0.

Revision notes and key takeaways

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

Showing 31-60 of 98 questions

Continue Your Practice

📚 Switch Set (ALL)

Chapter Sets

ALL 98 MCQs EASY 75 MCQs HARD 4 MCQs MEDIUM 19 MCQs

0 /98

0✓ 0✗

0% Accuracy