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Circle

278 Questions
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2022 JEE Mains MCQ
JEE Main 2022 (Online) 27th June Evening Shift

The set of values of k, for which the circle $C:4{x^2} + 4{y^2} - 12x + 8y + k = 0$ lies inside the fourth quadrant and the point $\left( {1, - {1 \over 3}} \right)$ lies on or inside the circle C, is :

A.
an empty set
B.
$\left( {6,{{65} \over 9}} \right]$
C.
$\left[ {{{80} \over 9},10} \right)$
D.
$\left( {9,{{92} \over 9}} \right]$
2022 JEE Mains MCQ
JEE Main 2022 (Online) 26th June Morning Shift

Let C be a circle passing through the points A(2, $-$1) and B(3, 4). The line segment AB s not a diameter of C. If r is the radius of C and its centre lies on the circle ${(x - 5)^2} + {(y - 1)^2} = {{13} \over 2}$, then r2 is equal to :

A.
32
B.
${{65} \over 2}$
C.
${{61} \over 2}$
D.
30
2022 JEE Mains MCQ
JEE Main 2022 (Online) 25th June Evening Shift

A circle touches both the y-axis and the line x + y = 0. Then the locus of its center is :

A.
$y = \sqrt 2 x$
B.
$x = \sqrt 2 y$
C.
${y^2} - {x^2} = 2xy$
D.
${x^2} - {y^2} = 2xy$
2022 JEE Mains MCQ
JEE Main 2022 (Online) 25th June Morning Shift
Let a circle C touch the lines ${L_1}:4x - 3y + {K_1} = 0$ and ${L_2} = 4x - 3y + {K_2} = 0$, ${K_1},{K_2} \in R$. If a line passing through the centre of the circle C intersects L1 at $( - 1,2)$ and L2 at $(3, - 6)$, then the equation of the circle C is :
A.
${(x - 1)^2} + {(y - 2)^2} = 4$
B.
${(x + 1)^2} + {(y - 2)^2} = 4$
C.
${(x - 1)^2} + {(y + 2)^2} = 16$
D.
${(x - 1)^2} + {(y - 2)^2} = 16$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th August Evening Shift
Let Z be the set of all integers,

$A = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {y^2} \le 4\} $

$B = \{ (x,y) \in Z \times Z:{x^2} + {y^2} \le 4\} $

$C = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {(y - 2)^2} \le 4\} $

If the total number of relation from A $\cap$ B to A $\cap$ C is 2p, then the value of p is :
A.
16
B.
25
C.
49
D.
9
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th August Evening Shift
A circle C touches the line x = 2y at the point (2, 1) and intersects the circle

C1 : x2 + y2 + 2y $-$ 5 = 0 at two points P and Q such that PQ is a diameter of C1. Then the diameter of C is :
A.
$7\sqrt 5 $
B.
15
C.
$\sqrt {285} $
D.
$4\sqrt {15} $
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th August Morning Shift
If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point ($-$30, 0) and is tangent to the parabola y2 = 30x, then the length of this chord is :
A.
5
B.
7
C.
5${\sqrt 3 }$
D.
3${\sqrt 5 }$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th July Evening Shift
Consider a circle C which touches the y-axis at (0, 6) and cuts off an intercept $6\sqrt 5 $ on the x-axis. Then the radius of the circle C is equal to :
A.
$\sqrt {53} $
B.
9
C.
8
D.
$\sqrt {82} $
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th July Morning Shift
Two tangents are drawn from the point P($-$1, 1) to the circle x2 + y2 $-$ 2x $-$ 6y + 6 = 0. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to :
A.
2
B.
$(3\sqrt 2 + 2)$
C.
4
D.
$3(\sqrt 2 - 1)$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th July Morning Shift
Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set {P, Q} is equal to :
A.
{(4, 0), (0, 6)}
B.
$\{ (2 + 2\sqrt 2 ,3 - \sqrt 5 ),(2 - 2\sqrt 2 ,3 + \sqrt 5 )\} $
C.
$\{ (2 + 2\sqrt 2 ,3 + \sqrt 5 ),(2 - 2\sqrt 2 ,3 - \sqrt 5 )\} $
D.
{($-$1, 5), (5, 1)}
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th July Morning Shift
Let $A = \{ (x,y) \in R \times R|2{x^2} + 2{y^2} - 2x - 2y = 1\} $, $B = \{ (x,y) \in R \times R|4{x^2} + 4{y^2} - 16y + 7 = 0\} $ and $C = \{ (x,y) \in R \times R|{x^2} + {y^2} - 4x - 2y + 5 \le {r^2}\} $.

Then the minimum value of |r| such that $A \cup B \subseteq C$ is equal to
A.
${{3 + \sqrt {10} } \over 2}$
B.
${{2 + \sqrt {10} } \over 2}$
C.
${{3 + 2\sqrt 5 } \over 2}$
D.
$1 + \sqrt 5 $
2021 JEE Mains MCQ
JEE Main 2021 (Online) 22th July Evening Shift
Let the circle S : 36x2 + 36y2 $-$ 108x + 120y + C = 0 be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, x $-$ 2y = 4 and 2x $-$ y = 5 lies inside the circle S, then :
A.
${{25} \over 9} < C < {{13} \over 3}$
B.
100 < C < 165
C.
81 < C < 156
D.
100 < C < 156
2021 JEE Mains MCQ
JEE Main 2021 (Online) 20th July Evening Shift
Let r1 and r2 be the radii of the largest and smallest circles, respectively, which pass through the point ($-$4, 1) and having their centres on the circumference of the circle x2 + y2 + 2x + 4y $-$ 4 = 0. If ${{{r_1}} \over {{r_2}}} = a + b\sqrt 2 $, then a + b is equal to :
A.
3
B.
11
C.
5
D.
7
2021 JEE Mains MCQ
JEE Main 2021 (Online) 18th March Evening Shift
Let S1 : x2 + y2 = 9 and S2 : (x $-$ 2)2 + y2 = 1. Then the locus of center of a variable circle S which touches S1 internally and S2 externally always passes through the points :
A.
$\left( {{1 \over 2}, \pm {{\sqrt 5 } \over 2}} \right)$
B.
(1, $\pm$ 2)
C.
$\left( {2, \pm {3 \over 2}} \right)$
D.
(0, $\pm$ $\sqrt 3 $)
2021 JEE Mains MCQ
JEE Main 2021 (Online) 18th March Morning Shift
Choose the correct statement about two circles whose equations are given below :

x2 + y2 $-$ 10x $-$ 10y + 41 = 0

x2 + y2 $-$ 22x $-$ 10y + 137 = 0
A.
circles have same centre
B.
circles have no meeting point
C.
circles have only one meeting point
D.
circles have two meeting points
2021 JEE Mains MCQ
JEE Main 2021 (Online) 18th March Morning Shift
For the four circles M, N, O and P, following four equations are given :

Circle M : x2 + y2 = 1

Circle N : x2 + y2 $-$ 2x = 0

Circle O : x2 + y2 $-$ 2x $-$ 2y + 1 = 0

Circle P : x2 + y2 $-$ 2y = 0

If the centre of circle M is joined with centre of the circle N, further center of circle N is joined with centre of the circle O, centre of circle O is joined with the centre of circle P and lastly, centre of circle P is joined with centre of circle M, then these lines form the sides of a :
A.
Rhombus
B.
Square
C.
Rectangle
D.
Parallelogram
2021 JEE Mains MCQ
JEE Main 2021 (Online) 17th March Evening Shift
Let the tangent to the circle x2 + y2 = 25 at the point R(3, 4) meet x-axis and y-axis at points P and Q, respectively. If r is the radius of the circle passing through the origin O and having centre at the incentre of the triangle OPQ, then r2 is equal to :
A.
${{585} \over {66}}$
B.
${{625} \over {72}}$
C.
${{529} \over {64}}$
D.
${{125} \over {72}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 17th March Evening Shift
Two tangents are drawn from a point P to the circle x2 + y2 $-$ 2x $-$ 4y + 4 = 0, such that the angle between these tangents is ${\tan ^{ - 1}}\left( {{{12} \over 5}} \right)$, where ${\tan ^{ - 1}}\left( {{{12} \over 5}} \right)$ $\in$(0, $\pi$). If the centre of the circle is denoted by C and these tangents touch the circle at points A and B, then the ratio of the areas of $\Delta$PAB and $\Delta$CAB is :
A.
3 : 1
B.
9 : 4
C.
2 : 1
D.
11 : 4
2021 JEE Mains MCQ
JEE Main 2021 (Online) 17th March Morning Shift
The line 2x $-$ y + 1 = 0 is a tangent to the circle at the point (2, 5) and the centre of the circle lies on x $-$ 2y = 4. Then, the radius of the circle is :
A.
5$\sqrt 3 $
B.
4$\sqrt 5 $
C.
3$\sqrt 5 $
D.
5$\sqrt 4 $
2021 JEE Mains MCQ
JEE Main 2021 (Online) 17th March Morning Shift
Choose the incorrect statement about the two circles whose equations are given below :

x2 + y2 $-$ 10x $-$ 10y + 41 = 0 and

x2 + y2 $-$ 16x $-$ 10y + 80 = 0
A.
Distance between two centres is the average of radii of both the circles.
B.
Both circles pass through the centre of each other.
C.
Circles have two intersection points.
D.
Both circle's centers lie inside region of one another.
2021 JEE Mains MCQ
JEE Main 2021 (Online) 16th March Evening Shift
Let the lengths of intercepts on x-axis and y-axis made by the circle
x2 + y2 + ax + 2ay + c = 0, (a < 0) be 2${\sqrt 2 }$ and 2${\sqrt 5 }$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line x + 2y = 0, is equal to :
A.
${\sqrt {10} }$
B.
${\sqrt {6} }$
C.
${\sqrt {11} }$
D.
${\sqrt {7} }$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Evening Shift
Let A(1, 4) and B(1, $-$5) be two points. Let P be a point on the circle
(x $-$ 1)2 + (y $-$ 1)2 = 1 such that (PA)2 + (PB)2 have maximum value, then the points, P, A and B lie on :
A.
a straight line
B.
an ellipse
C.
a parabola
D.
a hyperbola
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Evening Shift
If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x2 + y2 = 1 is a circle of radius r, then r is equal to :
A.
${1 \over 4}$
B.
${1 \over 2}$
C.
1
D.
${1 \over 3}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Morning Shift
In the circle given below, let OA = 1 unit, OB = 13 unit and PQ $ \bot $ OB. Then, the area of the triangle PQB (in square units) is :

JEE Main 2021 (Online) 26th February Morning Shift Mathematics - Circle Question 105 English
A.
24$\sqrt 2 $
B.
24$\sqrt 3 $
C.
26$\sqrt 2 $
D.
26$\sqrt 3 $
2020 JEE Mains MCQ
JEE Main 2020 (Online) 5th September Evening Slot
If the length of the chord of the circle,
x2 + y2 = r2 (r > 0) along the line, y – 2x = 3 is r,
then r2 is equal to :
A.
${9 \over 5}$
B.
${{24} \over 5}$
C.
${{12} \over 5}$
D.
12
2020 JEE Mains MCQ
JEE Main 2020 (Online) 4th September Evening Slot
The circle passing through the intersection of the circles,
x2 + y2 – 6x = 0 and x2 + y2 – 4y = 0, having its centre on
the line, 2x – 3y + 12 = 0, also passes through the point :
A.
(–3, 1)
B.
(1, –3)
C.
(–1, 3)
D.
(–3, 6)
2020 JEE Mains MCQ
JEE Main 2020 (Online) 9th January Morning Slot
A circle touches the y-axis at the point (0, 4) and passes through the point (2, 0). Which of the following lines is not a tangent to this circle?
A.
3x – 4y – 24 = 0
B.
4x + 3y – 8 = 0
C.
3x + 4y – 6 = 0
D.
4x – 3y + 17 = 0
2020 JEE Mains MCQ
JEE Main 2020 (Online) 8th January Evening Slot
If a line, y = mx + c is a tangent to the circle, (x – 3)2 + y2 = 1 and it is perpendicular to a line L1, where L1 is the tangent to the circle, x2 + y2 = 1 at the point $\left( {{1 \over {\sqrt 2 }},{1 \over {\sqrt 2 }}} \right)$, then :
A.
c2 + 6c + 7 = 0
B.
c2 - 7c + 6 = 0
C.
c2 – 6c + 7 = 0
D.
c2 + 7c + 6 = 0
2020 JEE Mains MCQ
JEE Main 2020 (Online) 7th January Evening Slot
Let the tangents drawn from the origin to the circle,
x2 + y2 - 8x - 4y + 16 = 0 touch it at the points A and B. The (AB)2 is equal to :
A.
${{56} \over 5}$
B.
${{32} \over 5}$
C.
${{52} \over 5}$
D.
${{64} \over 5}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Evening Slot
A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the point :
A.
(1, 5)
B.
( 2, 3)
C.
(3, 5)
D.
(3, 10)
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Morning Slot
If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90o, then the length (in cm) of their common chord is :
A.
${{13} \over 5}$
B.
${{60} \over {13}}$
C.
${{120} \over {13}}$
D.
${{13} \over 2}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Evening Slot
The locus of the centres of the circles, which touch the circle, x2 + y2 = 1 externally, also touch the y-axis and lie in the first quadrant, is :
A.
$x = \sqrt {1 + 2y} ,y \ge 0$
B.
$y = \sqrt {1 + 2x} ,x \ge 0$
C.
$y = \sqrt {1 + 4x} ,x \ge 0$
D.
$x = \sqrt {1 + 4y} ,y \ge 0$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Morning Slot
If the circles x2 + y2 + 5Kx + 2y + K = 0 and 2(x2 + y2) + 2Kx + 3y –1 = 0, (K$ \in $R), intersect at the points P and Q, then the line 4x + 5y – K = 0 passes through P and Q, for :
A.
exactly two values of K
B.
no value of K
C.
exactly one value of K
D.
infinitely many values of K
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Morning Slot
The line x = y touches a circle at the point (1,1). If the circle also passes through the point (1, – 3), then its radius is :
A.
3
B.
2
C.
2$\sqrt 2 $
D.
3$\sqrt 2 $
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Evening Slot
A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (–8, 5) and (6, 5), then the area of the rectangle (in sq. units) is :
A.
72
B.
84
C.
56
D.
98
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Evening Slot
The common tangent to the circles x 2 + y2 = 4 and x2 + y2 + 6x + 8y – 24 = 0 also passes through the point :
A.
(6, –2)
B.
(4, –2)
C.
(–4, 6)
D.
(–6, 4)
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Morning Slot
If a tangent to the circle x2 + y2 = 1 intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is :
A.
x2 + y2 – 4x2y2 = 0
B.
x2 + y2 - 2xy = 0
C.
x2 + y2 – 2x2y2 = 0
D.
x2 + y2 - 16x2y2 = 0
2019 JEE Mains MCQ
JEE Main 2019 (Online) 8th April Evening Slot
The tangent and the normal lines at the point ( $\sqrt 3 $, 1) to the circle x2 + y2 = 4 and the x-axis form a triangle. The area of this triangle (in square units) is :
A.
${4 \over {\sqrt 3 }}$
B.
${1 \over {\sqrt 3 }}$
C.
${2 \over {\sqrt 3 }}$
D.
${1 \over {3 }}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 8th April Morning Slot
The sum of the squares of the lengths of the chords intercepted on the circle, x2 + y2 = 16, by the lines, x + y = n, n $ \in $ N, where N is the set of all natural numbers, is :
A.
210
B.
160
C.
320
D.
105
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th January Evening Slot
If a circle of radius R passes through the origin O and intersects the coordinates axes at A and B, then the locus of the foot of perpendicular from O on AB is :
A.
(x2 + y2)2 = 4R2x2y2
B.
(x2 + y2) (x + y) = R2xy
C.
(x2 + y2)2 = 4Rx2y2
D.
(x2 + y2)3 = 4R2x2y2
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th January Morning Slot
If a variable line, 3x + 4y – $\lambda $ = 0 is such that the two circles x2 + y2 – 2x – 2y + 1 = 0 and x2 + y2 – 18x – 2y + 78 = 0 are on its opposite sides, then the set of all values of $\lambda $ is the interval :
A.
(23, 31)
B.
(2, 17)
C.
[13, 23]
D.
[12, 21]
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th January Morning Slot
Let C1 and C2 be the centres of the circles x2 + y2 – 2x – 2y – 2 = 0 and x2 + y2 – 6x – 6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral PC1QC2 is :
A.
4
B.
6
C.
9
D.
8
2019 JEE Mains MCQ
JEE Main 2019 (Online) 11th January Morning Slot
Two circles with equal radii are intersecting at the points (0, 1) and (0, –1). The tangent at the point (0, 1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is :
A.
$2\sqrt 2 $
B.
$\sqrt 2 $
C.
2
D.
1
2019 JEE Mains MCQ
JEE Main 2019 (Online) 11th January Morning Slot
A square is inscribed in the circle x2 + y2 – 6x + 8y – 103 = 0 with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is :
A.
$\sqrt {137} $
B.
6
C.
$\sqrt {41} $
D.
13
2019 JEE Mains MCQ
JEE Main 2019 (Online) 11th January Morning Slot
The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is :
A.
$4\sqrt 5 $
B.
${{\sqrt 5 } \over 2}$
C.
$2\sqrt 5 $
D.
${{\sqrt 5 } \over 4}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th January Evening Slot
If the area of an equilateral triangle inscribed in the circle x2 + y2 + 10x + 12y + c = 0 is $27\sqrt 3 $ sq units then c is equal to :
A.
20
B.
25
C.
$-$ 25
D.
13
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th January Morning Slot
If a circle C passing through the point (4, 0) touches the circle x2 + y2 + 4x – 6y = 12 externally at the point (1, – 1), then the radius of C is :
A.
5
B.
2$\sqrt {5} $
C.
4
D.
$\sqrt {37} $
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th January Evening Slot
If the circles

x2 + y2 $-$ 16x $-$ 20y + 164 = r2  

and  (x $-$ 4)2 + (y $-$ 7)2 = 36

intersect at two distinct points, then :
A.
r > 11
B.
0 < r < 1
C.
r = 11
D.
1 < r < 11
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th January Morning Slot
Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis as a common tangent, then :
A.
a, b, c are in A.P.
B.
$\sqrt a ,\sqrt b ,\sqrt c $ are in A.P
C.
${1 \over {\sqrt b }} + {1 \over {\sqrt c }}$ = ${1 \over {\sqrt a }}$
D.
${1 \over {\sqrt b }} = {1 \over {\sqrt a }} + {1 \over {\sqrt c }}$
2018 JEE Mains MCQ
JEE Main 2018 (Online) 16th April Morning Slot
If a circle C, whose radius is 3, touches externally the circle,
${x^2} + {y^2} + 2x - 4y - 4 = 0$ at the point (2, 2), then the length of the intercept cut by this circle C, on the x-axis is equal to :
A.
$2\sqrt 5 $
B.
$3\sqrt 2 $
C.
$\sqrt 5 $
D.
$2\sqrt 3 $