Circles · JEE Main previous year questions

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2025 15 questions

Q12025 · 8 Apr 2025 · Shift 2
Let the values of $λ$ for which the shortest distance between the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $\frac{x - λ}{3} = \frac{y - 4}{4} = \frac{z - 5}{5}$ is $\frac{1}{6}$ be $λ_{1}$ and $λ_{2}$ . Then the radius of the circle passing through the points $(0, 0)$ , $(λ_{1}, λ_{2})$ and $(λ_{2}, λ_{1})$ is
1. A. $3$
2. B. $\frac{5 2}{3}$
3. C. $\frac{2}{3}$
4. D. $4$
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Answer: B
Q72025 · 8 Apr 2025 · Shift 2
Let the ellipse $3 x^{2} + p y^{2} = 4$ pass through the centre $C$ of the circle $x^{2} + y^{2} - 2 x - 4 y - 11 = 0$ of radius $r$ . Let $f_{1}, f_{2}$ be the focal distances of the point $C$ on the ellipse. Then $6 f_{1} f_{2} - r$ is equal to
1. A. $78$
2. B. $68$
3. C. $70$
4. D. $74$
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Answer: C
Q232025 · 8 Apr 2025 · Shift 2· Numerical answer
Let $r$ be the radius of the circle, which touches $x$ -axis at point $(a, 0), a < 0$ and the parabola $y^{2} = 9 x$ at the point $(4, 6)$ . Then $r$ is equal to .
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Answer: 30
Q212025 · 4 Apr 2025 · Shift 1· Numerical answer
Let $C$ be the circle $x^{2} + (y - 1)^{2} = 2$ , $E_{1}$ and $E_{2}$ be two ellipses whose centres lie at the origin and major axes lie on x -axis and y -axis respectively. Let the straight line $x + y = 3$ touch the curves $C, E_{1}$ and $E_{2}$ at $P (x_{1}, y_{1}), Q (x_{2}, y_{2})$ and $R (x_{3}, y_{3})$ respectively. Given that $P$ is the mid point of the line segment $QR$ and $P Q = \frac{2 2}{3}$ , the value of $9 (x_{1} y_{1} + x_{2} y_{2} + x_{3} y_{3})$ is equal to _________.
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Answer: 46
Q62025 · 3 Apr 2025 · Shift 1
The radius of the smallest circle which touches the parabolas $y = x^{2} + 2$ and $x = y^{2} + 2$ is
1. A. $\frac{7 2}{16}$
2. B. $\frac{7 2}{8}$
3. C. $\frac{7 2}{2}$
4. D. $\frac{7 2}{4}$
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Answer: B
Q202025 · 3 Apr 2025 · Shift 2
If the four distinct points $(4, 6), (- 1, 5), (0, 0)$ and $(k, 3 k)$ lie on a circle of radius $r$ , then $10 k + r^{2}$ is equal to
1. A. $34$
2. B. $32$
3. C. $35$
4. D. $33$
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Answer: C
Q242025 · 2 Apr 2025 · Shift 1· Numerical answer
The absolute difference between the squares of the radii of the two circles passing through the point $(- 9, 4)$ and touching the lines $x + y = 3$ and $x - y = 3$ , is equal to _________ .
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Answer: 768
Q122025 · 29 Jan 2025 · Shift 2
Let a circle C pass through the points $(4, 2)$ and $(0, 2)$ , and its centre lie on $3 x + 2 y + 2 = 0$ . Then the length of the chord, of the circle C, whose mid-point is $(1, 2)$ , is:
1. A. $42$
2. B. $22$
3. C. $23$
4. D. $3$
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Answer: C
Q172025 · 28 Jan 2025 · Shift 1
Let the equation of the circle, which touches $x$ -axis at the point $(a, 0), a > 0$ and cuts off an intercept of length $b$ on $y$ -axis be $x^{2} + y^{2} - α x + β y + γ = 0$ . If the circle lies below $x$ -axis, then the ordered pair $(2 a, b^{2})$ is equal to
1. A. $(α, β^{2} + 4 γ)$
2. B. $(α, β^{2} - 4 γ)$
3. C. $(γ, β^{2} - 4 α)$
4. D. $(γ, β^{2} + 4 α)$
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Answer: B
Q202025 · 24 Jan 2025 · Shift 1
Let circle $C$ be the image of $x^{2} + y^{2} - 2 x + 4 y - 4 = 0$ in the line $2 x - 3 y + 5 = 0$ and $A$ be the point on $C$ such that $O A$ is parallel to $x$ -axis and $A$ lies on the right hand side of the centre $O$ of $C$ . If $B (α, β)$ , with $β < 4$ , lies on $C$ such that the length of the arc $A B$ is $(1/6)^{th}$ of the perimeter of $C$ , then $β - 3 α$ is equal to
1. A. $4 - 3$
2. B. $3$
3. C. $4$
4. D. $3 + 3$
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Answer: C
Q192025 · 23 Jan 2025 · Shift 2
Let the shortest distance from $(a, 0), a > 0$ , to the parabola $y^{2} = 4 x$ be $4$ . Then the equation of the circle passing through the point $(a, 0)$ and the focus of the parabola, and having its centre on the axis of the parabola is :
1. A. $x^{2} + y^{2} - 8 x + 7 = 0$
2. B. $x^{2} + y^{2} - 6 x + 5 = 0$
3. C. $x^{2} + y^{2} - 4 x + 3 = 0$
4. D. $x^{2} + y^{2} - 10 x + 9 = 0$
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Answer: B
Q222025 · 23 Jan 2025 · Shift 2· Numerical answer
The focus of the parabola $y^{2} = 4 x + 16$ is the centre of the circle $C$ of radius $5$ . If the values of $λ$ , for which $C$ passes through the point of intersection of the lines $3 x - y = 0$ and $x + λ y = 4$ , are $λ_{1}$ and $λ_{2}$ , $λ_{1} < λ_{2}$ , then $12 λ_{1} + 29 λ_{2}$ is equal to _________.
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Answer: 15
Q12025 · 22 Jan 2025 · Shift 1
The area of the region, inside the circle $(x - 23)^{2} + y^{2} = 12$ and outside the parabola $y^{2} = 23 x$ is :
1. A. $3 π - 8$
2. B. $6 π - 8$
3. C. $3 π + 8$
4. D. $6 π - 16$
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Answer: D
Q92025 · 22 Jan 2025 · Shift 1
A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point $(2, 5)$ and intersects the circle $C$ at exactly two points. If the set of all possible values of r is the interval $(α, β)$ , then $3 β - 2 α$ is equal to :
1. A.10
2. B.12
3. C.14
4. D.15
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Answer: D
Q172025 · 22 Jan 2025 · Shift 1
Let the parabola $y = x^{2} + p x - 3$ , meet the coordinate axes at the points $P, Q$ and $R$ . If the circle C with centre at $(- 1, - 1)$ passes through the points $P, Q$ and $R$ , then the area of $△ P QR$ is :
1. A.4
2. B.6
3. C.5
4. D.7
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Answer: B