Circles · JEE Main previous year questions

211 questions asked across 20 exam years, newest first

Year-wise papers Topic-wise questions

Practise this topic

2024 29 questions

Q42024 · 9 Apr 2024 · Shift 1
The parabola $y^{2} = 4 x$ divides the area of the circle $x^{2} + y^{2} = 5$ in two parts. The area of the smaller part is equal to :
1. A. $\frac{2}{3} + 5 sin^{- 1} (\frac{2}{5})$
2. B. $\frac{2}{3} + 5 sin^{- 1} (\frac{2}{5})$
3. C. $\frac{1}{3} + 5 sin^{- 1} (\frac{2}{5})$
4. D. $\frac{1}{3} + 5 sin^{- 1} (\frac{2}{5})$
Show answer
Answer: A
Q102024 · 9 Apr 2024 · Shift 1
Let a circle passing through $(2, 0)$ have its centre at the point $(h, k)$ . Let $(x_{c}, y_{c})$ be the point of intersection of the lines $3 x + 5 y = 1$ and $(2 + c) x + 5 c^{2} y = 1$ . If $h = lim_{c \to 1} x_{c}$ and $k = lim_{c \to 1} y_{c}$ , then the equation of the circle is :
1. A. $5 x^{2} + 5 y^{2} - 4 x - 2 y - 12 = 0$
2. B. $25 x^{2} + 25 y^{2} - 20 x + 2 y - 60 = 0$
3. C. $25 x^{2} + 25 y^{2} - 2 x + 2 y - 60 = 0$
4. D. $5 x^{2} + 5 y^{2} - 4 x + 2 y - 12 = 0$
Show answer
Answer: B
Q262024 · 9 Apr 2024 · Shift 1· Numerical answer
Let the centre of a circle, passing through the points $(0, 0)$ , $(1, 0)$ and touching the circle $x^{2} + y^{2} = 9$ , be $(h, k)$ . Then for all possible values of the coordinates of the centre $(h, k)$ , $4 (h^{2} + k^{2})$ is equal to ______.
Show answer
Answer: 9
Q52024 · 9 Apr 2024 · Shift 2
Two vertices of a triangle $ABC$ are $A (3, - 1)$ and $B (- 2, 3)$ , and its orthocentre is $P (1, 1)$ . If the coordinates of the point $C$ are $(α, β)$ and the centre of the of the circle circumscribing the triangle $PAB$ is $(h, k)$ , then the value of $(α + β) + 2 (h + k)$ equals
1. A. $81$
2. B. $15$
3. C. $51$
4. D. $5$
Show answer
Answer: D
Q252024 · 9 Apr 2024 · Shift 2· Numerical answer
Consider the circle $C : x^{2} + y^{2} = 4$ and the parabola $P : y^{2} = 8 x$ . If the set of all values of $α$ , for which three chords of the circle $C$ on three distinct lines passing through the point $(α, 0)$ are bisected by the parabola $P$ is the interval $(p, q)$ , then $(2 q - p)^{2}$ is equal to ____.
Show answer
Answer: 80
Q52024 · 8 Apr 2024 · Shift 1
Let the circles $C_{1} : (x - α)^{2} + (y - β)^{2} = r_{1}^{2}$ and $C_{2} : (x - 8)^{2} + (y - \frac{15}{2})^{2} = r_{2}^{2}$ touch each other externally at the point $(6, 6)$ . If the point $(6, 6)$ divides the line segment joining the centres of the circles $C_{1}$ and $C_{2}$ internally in the ratio $2 : 1$ , then $(α + β) + 4 (r_{1}^{2} + r_{2}^{2})$ equals
1. A. $130$
2. B. $110$
3. C. $145$
4. D. $125$
Show answer
Answer: A
Q42024 · 8 Apr 2024 · Shift 2
The area of the region in the first quadrant inside the circle $x^{2} + y^{2} = 8$ and outside the parabola $y^{2} = 2 x$ is equal to :
1. A. $\frac{π}{2} - \frac{1}{3}$
2. B. $π - \frac{1}{3}$
3. C. $π - \frac{2}{3}$
4. D. $\frac{π}{2} - \frac{2}{3}$
Show answer
Answer: C
Q192024 · 8 Apr 2024 · Shift 2
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:
1. A.2
2. B.3
3. C.4
4. D.1
Show answer
Answer: A
Q242024 · 8 Apr 2024 · Shift 2· Numerical answer
Let $S$ be the focus of the hyperbola $\frac{x ^{2}}{3} - \frac{y ^{2}}{5} = 1$ , on the positive $x$ -axis. Let $C$ be the circle with its centre at $A (6, 5)$ and passing through the point $S$ . If $O$ is the origin and $SAB$ is a diameter of $C$ , then the square of the area of the triangle $OSB$ is equal to ____.
Show answer
Answer: 40
Q132024 · 6 Apr 2024 · Shift 2
Suppose the solution of the differential equation $\frac{d y}{d x} = \frac{( 2 + α ) x - β y + 2}{β x - 2 α y - ( β γ - 4 α )}$ represents a circle passing through origin. Then the radius of this circle is :
1. A. $17$
2. B.2
3. C. $\frac{17}{2}$
4. D. $\frac{1}{2}$
Show answer
Answer: C
Q152024 · 6 Apr 2024 · Shift 2
If $P (6, 1)$ be the orthocentre of the triangle whose vertices are $A (5, - 2)$ , $B (8, 3)$ and $C (h, k)$ , then the point $C$ lies on the circle :
1. A. $x^{2} + y^{2} - 74 = 0$
2. B. $x^{2} + y^{2} - 65 = 0$
3. C. $x^{2} + y^{2} - 61 = 0$
4. D. $x^{2} + y^{2} - 52 = 0$
Show answer
Answer: B
Q12024 · 5 Apr 2024 · Shift 2
Let the circle $C_{1} : x^{2} + y^{2} - 2 (x + y) + 1 = 0$ and $C_{2}$ be a circle having centre at $(- 1, 0)$ and radius 2 . If the line of the common chord of $C_{1}$ and $C_{2}$ intersects the $y$ -axis at the point P, then the square of the distance of P from the centre of $C_{1}$ is:
1. A.4
2. B.6
3. C.2
4. D.1
Show answer
Answer: C
Q32024 · 5 Apr 2024 · Shift 2
Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies :
1. A. $r = 1$
2. B. $2 r^{2} - 4 r + 1 = 0$
3. C. $2 r^{2} - 8 r + 7 = 0$
4. D. $r^{2} - 8 r + 8 = 0$
Show answer
Answer: D
Q202024 · 5 Apr 2024 · Shift 2
The differential equation of the family of circles passing through the origin and having centre at the line $y = x$ is :
1. A. $(x^{2} - y^{2} + 2 x y) d x = (x^{2} - y^{2} + 2 x y) d y$
2. B. $(x^{2} + y^{2} - 2 x y) d x = (x^{2} + y^{2} + 2 x y) d y$
3. C. $(x^{2} + y^{2} + 2 x y) d x = (x^{2} + y^{2} - 2 x y) d y$
4. D. $(x^{2} - y^{2} + 2 x y) d x = (x^{2} - y^{2} - 2 x y) d y$
Show answer
Answer: D
Q302024 · 5 Apr 2024 · Shift 2· Numerical answer
Let a line perpendicular to the line $2 x - y = 10$ touch the parabola $y^{2} = 4 (x - 9)$ at the point P. The distance of the point P from the centre of the circle $x^{2} + y^{2} - 14 x - 8 y + 56 = 0$ is _________.
Show answer
Answer: 10
Q52024 · 4 Apr 2024 · Shift 1
A square is inscribed in the circle $x^{2} + y^{2} - 10 x - 6 y + 30 = 0$ . One side of this square is parallel to $y = x + 3$ . If $(x_{i}, y_{i})$ are the vertices of the square, then $\sum (x_{i}^{2} + y_{i}^{2})$ is equal to:
1. A.152
2. B.148
3. C.156
4. D.160
Show answer
Answer: A
Q142024 · 4 Apr 2024 · Shift 2
Let $C$ be a circle with radius $10$ units and centre at the origin. Let the line $x + y = 2$ intersect the circle $C$ at the points $P$ and $Q$ . Let $MN$ be a chord of $C$ of length 2 unit and slope $- 1$ . Then, a distance (in units) between the chord $PQ$ and the chord $MN$ is
1. A. $3 - 2$
2. B. $2 - 3$
3. C. $2 - 1$
4. D. $2 + 1$
Show answer
Answer: A
Q182024 · 1 Feb 2024 · Shift 1
Let $C : x^{2} + y^{2} = 4$ and $C^{'} : x^{2} + y^{2} - 4 λ x + 9 = 0$ be two circles. If the set of all values of $λ$ so that the circles $C$ and $C^{'}$ intersect at two distinct points, is $R - [a, b]$ , then the point $(8 a + 12, 16 b - 20)$ lies on the curve :
1. A. $x^{2} + 2 y^{2} - 5 x + 6 y = 3$
2. B. $5 x^{2} - y = - 11$
3. C. $x^{2} - 4 y^{2} = 7$
4. D. $6 x^{2} + y^{2} = 42$
Show answer
Answer: D
Q262024 · 1 Feb 2024 · Shift 1· Numerical answer
Let the line $L : 2 x + y = α$ pass through the point of the intersection $P$ (in the first quadrant) of the circle $x^{2} + y^{2} = 3$ and the parabola $x^{2} = 2 y$ . Let the line $L$ touch two circles $C_{1}$ and $C_{2}$ of equal radius $23$ . If the centres $Q_{1}$ and $Q_{2}$ of the circles $C_{1}$ and $C_{2}$ lie on the $y$ -axis, then the square of the area of the triangle $P Q_{1} Q_{2}$ is equal to ____.
Show answer
Answer: 72
Q132024 · 1 Feb 2024 · Shift 2
Let the locus of the midpoints of the chords of the circle $x^{2} + (y - 1)^{2} = 1$ drawn from the origin intersect the line $x + y = 1$ at $P$ and $Q$ . Then, the length of $PQ$ is :
1. A. $\frac{1}{2}$
2. B.1
3. C. $\frac{1}{2}$
4. D. $2$
Show answer
Answer: C
Q192024 · 31 Jan 2024 · Shift 1
If one of the diameters of the circle $x^{2} + y^{2} - 10 x + 4 y + 13 = 0$ is a chord of another circle $C$ , whose center is the point of intersection of the lines $2 x + 3 y = 12$ and $3 x - 2 y = 5$ , then the radius of the circle $C$ is :
1. A.4
2. B. $32$
3. C.6
4. D. $20$
Show answer
Answer: C
Q182024 · 31 Jan 2024 · Shift 2
Let a variable line passing through the centre of the circle $x^{2} + y^{2} - 16 x - 4 y = 0$ , meet the positive co-ordinate axes at the points $A$ and $B$ . Then the minimum value of $O A + O B$ , where $O$ is the origin, is equal to
1. A.12
2. B.20
3. C.24
4. D.18
Show answer
Answer: D
Q32024 · 30 Jan 2024 · Shift 1
If the circles $(x + 1)^{2} + (y + 2)^{2} = r^{2}$ and $x^{2} + y^{2} - 4 x - 4 y + 4 = 0$ intersect at exactly two distinct points, then
1. A. $\frac{1}{2} < r < 7$
2. B. $3 < r < 7$
3. C. $5 < r < 9$
4. D. $0 < r < 7$
Show answer
Answer: B
Q32024 · 29 Jan 2024 · Shift 1
Let $(5, \frac{a}{4})$ be the circumcenter of a triangle with vertices $A (a, - 2), B (a, 6)$ and $C (\frac{a}{4}, - 2)$ . Let $α$ denote the circumradius, $β$ denote the area and $γ$ denote the perimeter of the triangle. Then $α + β + γ$ is
1. A. $60$
2. B. $62$
3. C. $53$
4. D. $30$
Show answer
Answer: C
Q212024 · 29 Jan 2024 · Shift 1· Numerical answer
The area (in sq. units) of the part of the circle $x^{2} + y^{2} = 169$ which is below the line $5 x - y = 13$ is $\frac{π α}{2 β} - \frac{65}{2} + \frac{α}{β} sin^{- 1} (\frac{12}{13})$ , where $α, β$ are coprime numbers. Then $α + β$ is equal to _______.
Show answer
Answer: 171