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JEE • MATHEMATICS • CIRCLES

Circles practice · Hard

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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 Hard

1 of 16

Application +4 | -1

The sum of the coefficients in the expansion of

{\left( {{\alpha ^2}{x^2} - 2\alpha \mathrm{ }x + 1} \right)^{51}},

as a polynomial in

x

, vanishes. Position of the point

\left( {\alpha ,2{\alpha ^2}} \right)

with respect to the circle

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

View answer and explanation

Correct option: A

Outside

Option A (Outside) is correct because substituting

\alpha = 1

into the circle's equation shows that the point (1, 2) lies outside the circle.

2 Hard

2 of 16

Analyzing +4 | -1

The values of

\theta

\left[ {0,2\pi } \right]

so that circles

{x^2} + {y^2} + 2\left( {\sin \alpha } \right)x + 2\left( {\cos \alpha } \right)y + {\sin ^2}\theta = 0

always lies inside the square of unit side length, is /are

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the correct range for

\theta

\left[ {\pi /3,2\pi /3} \right] \cup \left[ {4\pi /3,5\pi /3} \right]

, which is not listed in the options.

3 Hard

3 of 16

Analyzing +4 | -1

The set of values of '

c

' so that the equation

y = \left| x \right| + c

and

{x^2} + {y^2} - 8\left| x \right| - 9 = 0

have no solution, is

View answer and explanation

Correct option: D

\left( {5\sqrt 2 - 4,\;\infty } \right)

Option C is correct because eliminating the parameter 't' results in the equation of a parabola.

4 Hard

4 of 16

Analyzing +4 | -1

If the circles

{x^2} + {y^2} + 2ax + cy + a = 0

and

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

intersect in two distinct points

P

and

Q

, then the line

5x + by - a = 0

passes through

P

and

Q

for

View answer and explanation

Correct option: C

No value of

a

Option C (No value of

a

) is correct because the condition for the line to pass through the intersection points leads to a quadratic equation with no real solutions for

a

5 Hard

5 of 16

Application +4 | -1

If the tangent at the point

P

on the circle

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

meets the straight line.

5x - 2y + 6 = 0

at a point

Q

on the

y

-axis, then the length of

PQ

View answer and explanation

Correct option: C

Option C (5) is correct because it represents the length of the tangent segment from the point Q on the y-axis to the circle.

6 Hard

6 of 16

Application +4 | -1

One of the diameters of the circle circumscribing the rectangle

ABCD

4y = x + 7.

A

and

B

are the points

\left( { - 3,4} \right)

and

\left( {5,4} \right)

respectively, then the area of the rectangle is

View answer and explanation

Correct option: C

32 sq. units

Option C (32 sq. units) is correct because it is derived from finding the center of the circle, the length of the rectangle, and the perpendicular distance to determine the width.

7 Hard

7 of 16

Application +4 | -1

The equation to the circle having

y = mx

as a diameter where

y = mx

is a chord of the circle, through the origin, of radius a and having the x-axis as diameter is

View answer and explanation

Correct option: A

\left( {1 + {m^2}} \right)\left( {{x^2} + {y^2}} \right) - 2a\left( {x + my} \right) = 0

The correct answer is

\left( {1 + {m^2}} \right)\left( {{x^2} + {y^2}} \right) - 2a\left( {x + my} \right) = 0

because it represents the equation of the circle with the given conditions.

8 Hard

8 of 16

Application +4 | -1

P\left( { - 1, - 3} \right)

is a centre of similitude for the circles

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

and

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

then the length of the common tangent through

P

to the circles is

View answer and explanation

Correct option: B

The correct answer is 3 because it represents the length of the common tangent through P to the circles, calculated using the lengths of tangents from P to each circle.

9 Hard

9 of 16

Application +4 | -1

If a point

P

has co-ordinates

(0,-2)

and

Q

is any point on the circle,

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

, then the maximum value of

(P Q)^{2}

is :

View answer and explanation

Correct option: B

14+5 \sqrt{3}

Option B (

14+5 \sqrt{3}

) is correct because it represents the maximum value of (PQ)² calculated by parameterizing the circle and maximizing the distance.

10 Hard

10 of 16

Analyzing +4 | -1

Three sides of a triangle have the equations

{L_r} \equiv y - {m_r}x - {c_r} = 0,r = 1,2,3.

\lambda ,\mu ,v

are non-zero real numbers such that

\lambda \;{L_2}{L_3} + \mu {L_3}{L_1} + v\;{L_1}{L_2} = 0

represents the circumcircle of the triangle, then

View answer and explanation

Correct option: C

Both (1) and (2) hold together

The correct answer is C. For the given equation to represent the circumcircle, it must satisfy the conditions for a circle, which leads to both equations in options A and B holding true.

11 Hard

11 of 16

Application +4 | -1

The equation of the circle radius

2\sqrt 2

whose centre lies on the line

x - y = 0

and which touches the line

x + y = 4,

and whose centre is coordinate satisfy

x + y > 4,

View answer and explanation

Correct option: A

{x^2} + {y^2} - 8x - 8y + 24 = 0

Option A (

{x^2} + {y^2} - 8x - 8y + 24 = 0

) is correct because the center of the circle lies on the line

x - y = 0

, touches the line

x + y = 4

, and the coordinates of the center satisfy

x + y > 4

12 Hard

12 of 16

Analyzing +4 | -1

If a variable tangent of the circle

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

intersects the ellipse

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

at points

P

and

Q

, then the locus of the point of intersection of tangent at

P

and

Q

View answer and explanation

Correct option: C

An ellipse with eccentricity

\frac{{\sqrt 3 }}{2}

The correct answer is C. The locus of the intersection point is an ellipse, and its eccentricity is calculated to be

\frac{{\sqrt 3 }}{{2}}

13 Hard

13 of 16

Analyzing +4 | -1

The two lines through

\left( {2,3} \right)

from which the circle

{x^2} + {y^2} = 25

intercepts chords of length 8 units have equations

View answer and explanation

Correct option: B

y = 3,12x + 5y = 39

Option B (

y = 3,12x + 5y = 39

) is correct because these lines satisfy the condition of intercepting chords of length 8 units from the given circle when passing through (2,3).

14 Hard

14 of 16

Application +4 | -1

Consider a family of circles, which are passing through the point

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

and are tangent to

x

-axis. If

\left( {h,\;k} \right)

are the coordinates of the centre of the circles, then the set of values of

k

is given by the interval

View answer and explanation

Correct option: B

k \ge \frac{1}{2}

Option B (

k \ge \frac{1}{2}

) is correct because it represents the range of possible y-coordinates for the center of the circle, derived from the condition that the circle is tangent to the x-axis and passes through (-1, 1).

15 Hard

15 of 16

Analyzing +4 | -1

Two concentric circles of which smallest is

x^2+y^2=4

, have the difference in radii as

d

, if line

y=x+1

cuts the circles in real points, then

d

lies in the interval

View answer and explanation

Correct option: A

\left(-\infty,-2-\frac{1}{\sqrt{2}}\right) \cup\left(-2+\frac{1}{\sqrt{2}}, \infty\right)

The correct answer is A

\left(-\infty,-2-\frac{{1}}{{\sqrt{{2}}}}}\right) \cup\left(-2+\frac{{1}}{{\sqrt{{2}}}}, \infty\right)

because it represents the interval for the difference in radii such that the line intersects both circles in real points.

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16 Hard

16 of 16

Application +4 | -1

The equation of the circle which passes through the intersection of

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

and

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

and whose centre lies on

13x + 30y = 0

, is

View answer and explanation

Correct option: B

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

Option C (

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

) is correct because it represents the equation of a circle passing through the intersection of two given circles and whose center lies on the given line.

Revision notes and key takeaways

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

Key Takeaways

✓ This set contains 16 single-correct-answer MCQs on 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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