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JEE • MATHEMATICS • CIRCLES

Circles practice · Easy

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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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Showing 1-30 of 480 questions

1 Easy

1 of 480

Application +4 | -1

One of the diameter of the circle

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

View answer and explanation

Correct option: A

x - y - 3 = 0

Option A (

x - y - 3 = 0

) is correct because the diameter of a circle passes through its center, and this line equation satisfies the center's coordinates.

2 Easy

2 of 480

Analyzing +4 | -1

\quad \alpha, \beta

and

\gamma

are the parametric angles of three points

P, Q

, and R respectively, on the circle

x^2+y^2=1

, and

A

is the point

(-1,0)

. If the lengths of the chords

A P, A Q

and

A R

are in

G P

, then

\cos \alpha / 2, \cos \beta / 2

and

\cos \gamma / 2

are in

View answer and explanation

Correct option: B

Option B (GP) is correct because the lengths of the chords AP, AQ, and AR being in GP implies that cos(alpha/2), cos(beta/2), and cos(gamma/2) are also in GP.

3 Easy

3 of 480

Application +4 | -1

The radius of the circle passing through the point (6,2), two of whose diameters are

x+y=6

and

x+2 y=4

View answer and explanation

Correct option: B

2 \sqrt{5}

Option B (

2 \sqrt{5}

) is correct because it is the distance between the center of the circle (intersection of diameters) and a point on the circle (6,2).

4 Easy

4 of 480

Application +4 | -1

If a circle has two of its diameters along the lines

x+y=5

and

x-y=1

and has area

9 \pi

, then the equation of the circle is

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because the center is the intersection of the diameters, and the radius is derived from the area.

5 Easy

5 of 480

Analyzing +4 | -1

Four distinct points

\left( {2k,3k} \right),\left( {1,0} \right),\left( {0,1} \right)

and

\left( {0,0} \right)

lie on a circle for

View answer and explanation

Correct option: D

For two values of

k

Option D (For two values of

k

) is correct because there are two non-trivial values of

k

for which the four points lie on a circle.

6 Easy

6 of 480

Application +4 | -1

The radius of the circle whose arc of length

15\,\pi \,\,\mathrm{cm}

makes an angle of

\frac{{3\,\pi }}{4}

radian at the centre is

View answer and explanation

Correct option: B

20\,\,\mathrm{cm}

Option B (20 cm) is correct because the radius is calculated using the formula relating arc length, angle, and radius: arc length = radius * angle.

7 Easy

7 of 480

Application +4 | -1

Equation of the tangent to the circle at the point

(1,-1)

whose centre is the point of intersection of the straight lines

x-y=1

and

2 x+y-3=0

, is

View answer and explanation

Correct option: B

x+4 y+3=0

The correct answer is

x+4 y+3=0

because it is derived by substituting the point and center into the tangent equation.

8 Easy

8 of 480

Application +4 | -1

The equation of the image of the circle

{x^2} + {y^2} + 16x - 24y + 183 = 0

by the line mirror

4x + 7y + 13 = 0

View answer and explanation

Correct option: D

{x^2} + {y^2} + 32x + 4y + 235 = 0

Option D (

{x^2} + {y^2} + 32x + 4y + 235 = 0

) is correct because it represents the image of the given circle after reflection across the line, found by finding the image of the center.

9 Easy

9 of 480

Application +4 | -1

If the image of the point

(-4,5)

in the line

x+2 y=2

lies on the circle

(x+4)^2+(y-3)^2=r^2,

then

r

is equal to:

View answer and explanation

Correct option: B

Option B (2) is correct because by finding the image of the point (-4,5) in the line x+2y=2 and substituting it into the circle's equation, we can solve for r and find that r=2.

10 Easy

10 of 480

Application +4 | -1

The radius of the circle touching the straight lines

x-2 y-1=0

and

3 x-6 y+7=0

, is

View answer and explanation

Correct option: B

\frac{\sqrt{5}}{3}

unit

Option A (

\frac{\sqrt{5}}{3}

unit) is correct because the radius is half the distance between the two parallel lines.

11 Easy

11 of 480

Application +4 | -1

If the lines

2 x+3 y+1=0

and

3 x-y-4=0

lie along diameters of a circle of circumference

10 \pi

, then the equation of the circle is

View answer and explanation

Correct option: D

x^2+y^2-2 x+2 y-23=0

Option D (

x^2+y^2-2 x+2 y-23=0

) is correct because the center is the intersection of the diameters, and the radius is derived from the circumference.

12 Easy

12 of 480

Application +4 | -1

The locus of the centre of a circle touching the lines

x+2 y=0

and

x-2 y=0

, is

View answer and explanation

Correct option: A

x y=0

Option A (xy=0) is correct because the locus of the center is found to be xy=0, representing the x and y axes.

13 Easy

13 of 480

Application +4 | -1

The number of points

(x, y)

having integral coordinates satisfying the condition

x^2+y^2<25

View answer and explanation

Correct option: A

Option A (69) is correct because after considering the integral coordinates within the circle

x^2 + y^2 < 25

, and excluding the invalid cases, the total number of points is 69.

14 Easy

14 of 480

Application +4 | -1

An equation of the normal at the point

(2,3)

to the circle

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

View answer and explanation

Correct option: C

2 x-y-1=0

Option C (

2 x-y-1=0

) is correct because the normal to a circle at a point passes through the center of the circle.

15 Easy

15 of 480

Application +4 | -1

The coordinates of the point on the circle

x^2+y^2-2 x-4 y-11=0

farthest from the origin are

View answer and explanation

Correct option: B

(1+4 / \sqrt{5}, 2+8 / \sqrt{5})

Option B (

(1+4 / \sqrt{5}, 2+8 / \sqrt{5})

) is correct because the point farthest from the origin lies on the line connecting the origin and the center of the circle, and it is at a distance of radius from the center.

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16 Easy

16 of 480

Application +4 | -1

The equations of the tangents to circle

{x^2} + {y^2} - 6x + 4y - 12 = 0

, which are parallel to line

4x + 3y + 15

are

View answer and explanation

Correct option: C

4x + 3y + 19 = 0

and

4x + 3y - 31 = 0

Option C (

4x + 3y + 19 = 0

and

4x + 3y - 31 = 0

) is correct because these are the equations of the tangents to the circle that are parallel to the given line.

17 Easy

17 of 480

Application +4 | -1

The locus of the middle points of chords of the circle

x^2+y^2=25

which are parallel to the line

x-2 y+3=0

, is

View answer and explanation

Correct option: B

2 x+y=0

Option B (

2 x+y=0

) is correct because it represents the locus of the midpoints of the chords parallel to the given line, derived from the properties of circles and lines.

18 Easy

18 of 480

Analyzing +4 | -1

For the given circles

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

and

{x^2} + {y^2} + 2x - 8y + 13 = 0

, which of the following is true?

View answer and explanation

Correct option: D

They touch each other externally

The correct answer is D. By calculating the distance between the centers and comparing it to the sum of the radii, it's found that the circles touch each other externally.

19 Easy

19 of 480

Application +4 | -1

The number of integral values of

\lambda

for which

x^2+y^2+\lambda x+(1-\lambda) y+5=0

is the equation of a circle whose radius cannot exceed 5 , is

View answer and explanation

Correct option: C

Option C (16) is correct because the number of integral values of

\lambda

that satisfy the condition for the radius of the circle not exceeding 5 is 16.

20 Easy

20 of 480

Application +4 | -1

Area of the triangle formed by tangent of the curve

\mathrm{x}=\mathrm{t}^2+\mathrm{t}+2, \mathrm{y}=\mathrm{t}^2-\mathrm{t}+2, \mathrm{t} \geq 0

which passes through

(2,2)

and co-ordinate axes is equal to

View answer and explanation

Correct option: D

Option D (8) is correct because the area of the triangle formed by the tangent to the curve at (2,2) and the coordinate axes is calculated to be 8.

21 Easy

21 of 480

Application +4 | -1

If the radii of the circles

(x-1)^2+(y-2)^2=1

and

(x-7)^2+(y-10)^2=4

are increasing uniformly w.r.t. time as

0.3 \mathrm{\,unit} / \mathrm{s}

and

0.4 \mathrm{\,unit} / \mathrm{s}

respectively, then they will touch each other at

t

equals to

View answer and explanation

Correct option: B

90 \mathrm{~s}

Option B (90 s) is correct because it represents the time at which the distance between the centers equals the sum of the radii.

22 Easy

22 of 480

Application +4 | -1

The angular between the tangent drawn from the origin to the circle

{\left( {x - 7} \right)^2} + {\left( {y + 1} \right)^2} = 25

View answer and explanation

Correct option: C

\frac{\pi }{2}

Option C (

\frac{\pi}{2}

) is correct because the product of the slopes of the tangents from the origin to the circle is -1, indicating perpendicularity.

23 Easy

23 of 480

Application +4 | -1

Circles are drawn through the point

(3,0)

to cut an intercept of length 6 units on the negative direction of the

x

-axis. The equation of the locus of their centres, is

View answer and explanation

Correct option: C

x=0

Option C (x=0) is correct because the locus of the centers of the circles is found to be x=0.

24 Easy

24 of 480

Application +4 | -1

If the straight line

x - 2y + 1 = 0

intersects the circle

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

in points

P

and

Q

, then the coordinates of the point of intersection of tangents drawn at

P

and

Q

to the circle

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

are

View answer and explanation

Correct option: C

\left( { - 25,50} \right)

Option C ((-25,50)) is correct because it represents the point of intersection of the tangents to the circle at points P and Q, where the line x - 2y + 1 = 0 intersects the circle.

25 Easy

25 of 480

Application +4 | -1

If a square with length of each side equal to

a

is inscribed in the circle

x^2+y^2+4 x+10 y+21=0

, then

a

is equal to

View answer and explanation

Correct option: D

Option D (4) is correct because the side length of the inscribed square is calculated using the circle's radius and the relationship between the square's diagonal and side.

26 Easy

26 of 480

Analyzing +4 | -1

The number of common tangents to the circles

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

View answer and explanation

Correct option: D

The correct answer is 3 because the circles touch each other externally, resulting in three common tangents: one direct common tangent and two transverse common tangents.

27 Easy

27 of 480

Application +4 | -1

If the lengths of the tangents from two points

A

and

B

to a circle are

l

and

l^{\prime}

respectively and points are conjugate with respect to the circle (i.e. each passes through the polar of the other with respect to the circle), then

(A B)^2=

View answer and explanation

Correct option: C

l^2+l^2

Option C (

l^2+l^{\prime 2}

) is correct because it represents the square of the distance between two conjugate points with respect to a circle.

28 Easy

28 of 480

Application +4 | -1

The value of

k

for which two tangents can be drawn from

(k, k)

to the circle

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

View answer and explanation

Correct option: C

k \in(-\infty,-4) \cup(2, \infty)

Option C,

k \in(-\infty,-4) \cup(2, \infty)

, is correct because it represents the values of k for which the point (k, k) lies outside the circle, allowing two tangents to be drawn.

29 Easy

29 of 480

Application +4 | -1

The locus of the point of intersection of perpendicular tangents to the circles

x^2+y^2=a^2

and

x^2+y^2=b^2

, is

View answer and explanation

Correct option: B

x^2+y^2=a^2+b^2

The correct answer is

x^2+y^2=a^2+b^2

because the locus of the intersection of perpendicular tangents is the sum of the squares of the radii.

30 Easy

30 of 480

Application +4 | -1

The locus of a point such that the tangents drawn from it to the circle

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

are perpendicular to each other is

View answer and explanation

Correct option: A

x^2+y^2-6 x-8 y-25=0

Option A (

x^2+y^2-6 x-8 y-25=0

) is correct because the locus of the point of intersection of perpendicular tangents to a circle is the director circle, which has the same center as the original circle and a radius of

r\sqrt{{2}}

Revision notes and key takeaways

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Key Takeaways

✓ This set contains 480 single-correct-answer MCQs on CIRCLES.
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