Circles · JEE Main previous year questions

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2026 25 questions

Q242026 · 8 Apr 2026 · Shift 2· Numerical answer
Consider the circle $C : x^{2} + y^{2} - 6 x - 8 y - 11 = 0$ . Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle $x^{2} + y^{2} - α x - β y - γ = 0$ , then $α + β + 2 γ$ is equal to ____ .
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Answer: 18
Q222026 · 6 Apr 2026 · Shift 1· Numerical answer
Let the centre of the circle $x^{2} + y^{2} + 2 g x + 2 f y + 25 = 0$ be in the first quadrant and lie on the line $2 x - y = 4$ . Let the area of an equilateral triangle inscribed in the circle be $273$ . Then the square of the length of the chord of the circle on the line $x = 1$ is ____.
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Answer: 80
Q102026 · 6 Apr 2026 · Shift 2
Let C be a circle having centre in the first quadrant and touching the $x$ -axis at a distance of 3 units from the origin. If the circle C has an intercept of length $63$ on $y$ -axis, then the length of the chord of the circle C on the line $x - y = 3$ is :
1. A. $8$
2. B. $6$
3. C. $62$
4. D. $82$
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Answer: C
Q232026 · 6 Apr 2026 · Shift 2· Numerical answer
Let the line $x - y = 4$ intersect the circle $C : (x - 4)^{2} + (y + 3)^{2} = 9$ at the points $Q$ and $R$ . If $P (α, β)$ is a point on $C$ such that $PQ = PR$ , then $(6 α + 8 β)^{2}$ is equal to ____ .
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Answer: 18
Q102026 · 5 Apr 2026 · Shift 1
1. A.8
2. B.10
3. C.12
4. D.9
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Answer: C
Q252026 · 5 Apr 2026 · Shift 1· Numerical answer
Let $y = y (x)$ be the solution of the differential equation $x sin (\frac{y}{x}) d y = (y sin (\frac{y}{x}) - x) d x, y (1) = \frac{π}{2} and let α = cos (\frac{y ( e ^{12} )}{e ^{12}}) .$ Then the number of integral value of $p$ , for which the equation $x^{2} + y^{2} - 2 p x + 2 p y + α + 2 = 0$ represents a circle of radius $r \leq 6$ , is ____.
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Answer: 0 to 1000000000
Q112026 · 5 Apr 2026 · Shift 2
Let the point $P$ be the vertex of the parabola $y = x^{2} - 6 x + 12$ . If a line passing through the point $P$ intersects the circle $x^{2} + y^{2} - 2 x - 4 y + 3 = 0$ at the points $R$ and $S$ , then the maximum value of $(PR + PS)^{2}$ is :
1. A. $10$
2. B. $20$
3. C. $25$
4. D. $5$
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Answer: B
Q132026 · 4 Apr 2026 · Shift 1
Suppose that two chords, drawn from the point $(1, 2)$ on the circle $x^{2} + y^{2} + x - 3 y = 0$ are bisected by the $y$ -axis. If the other ends of these chords are R and S, and the mid point of the line segment RS is $(α, β)$ , then $6 (α + β)$ is equal to:
1. A.1
2. B.3
3. C.4
4. D.6
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Answer: B
Q62026 · 4 Apr 2026 · Shift 2
Let $A = 143 2 - 2 8 78 - 7$ and $det (A - α I) = 0$ , where $α$ is a real number. If the largest possible value of $α$ is $p$ , then the circle $(x - p)^{2} + (y - 2 p)^{2} = 320$ , intersects the co-ordinate axes at
1. A.1 point
2. B.2 points
3. C.3 points
4. D.4 points
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Answer: C
Q102026 · 4 Apr 2026 · Shift 2
Let $P (3 cos α, 2 sin α)$ , $α \neq = 0$ , be a point on the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ , Q be a point on the circle $x^{2} + y^{2} - 14 x - 14 y + 82 = 0$ and R be a point on the line $x + y = 5$ such that the centroid of the triangle PQR is $(2 + cos α, 3 + \frac{2}{3} sin α)$ . Then the sum of the ordinates of all possible points R is:
1. A.6
2. B.2
3. C.4
4. D.8
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Answer: D
Q232026 · 4 Apr 2026 · Shift 2· Numerical answer
Let A, B be points on the two half-lines $x - 3 ∣ y ∣ = α$ , $α > 0$ at a distance of $α$ from their point of intersection P. The line segment AB meets the angle bisector of the given half-lines at the point Q. If $PQ = \frac{9}{2}$ and R is the radius of the circumcircle of $Δ PAB$ , then $\frac{α ^{2}}{R}$ is equal to ____.
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Answer: 9
Q242026 · 2 Apr 2026 · Shift 1· Numerical answer
Let a circle $C$ have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of $C$ on the line $x + y = 1$ is $14$ , then the square of the radius of $C$ is ____.
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Answer: 8
Q22026 · 2 Apr 2026 · Shift 2
Let the circles $C_{1} : ∣ z ∣ = r$ and $C_{2} : ∣ z - 3 - 4 i ∣ = 5$ , $z \in C$ , be such that $C_{2}$ lies within $C_{1}$ . If $z_{1}$ moves on $C_{1}$ , $z_{2}$ moves on $C_{2}$ and $min ∣ z_{1} - z_{2} ∣ = 2$ , then $max ∣ z_{1} - z_{2} ∣$ is equal to :
1. A.12
2. B.17
3. C.22
4. D.24
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Answer: C
Q102026 · 2 Apr 2026 · Shift 2
Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines $x + (k - 1) y + 3 = 0$ and $2 x + k^{2} y - 4 = 0$ . If the line $x - y + 2 = 0$ intersects the circle at the points A and B, then $(AB)^{2}$ is equal to :
1. A.10
2. B.27
3. C.18
4. D.34
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Answer: C
Q232026 · 2 Apr 2026 · Shift 2· Numerical answer
Let A be the point $(3, 0)$ and circles with variable diameter AB touch the circle $x^{2} + y^{2} = 36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is $e$ , then $72 e^{2}$ is equal to ____.
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Answer: 18
Q62026 · 28 Jan 2026 · Shift 1
Let $y = x$ be the equation of a chord of the circle $C_{1}$ (in the closed half-plane $x \geq 0$ ) of diameter $10$ passing through the origin. Let $C_{2}$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $C_{2}$ , which passes through the point $(2, 3)$ and is farthest from the center of $C_{2}$ , is $x + a y + b = 0$ , then $a - b$ is equal to
1. A.-6
2. B.10
3. C.6
4. D.-2
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Answer: D
Q92026 · 28 Jan 2026 · Shift 1
Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line $x + 22 y = 4$ . If the co-ordinates of the vertex A are $(α, β)$ , then the greatest integer less than or equal to $∣ α + 2 β ∣$ is
1. A.5
2. B.4
3. C.2
4. D.3
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Answer: B
Q12026 · 28 Jan 2026 · Shift 2
Let the circle $x^{2} + y^{2} = 4$ intersect x-axis at the points $A (a, 0), a > 0$ and $B (b, 0)$ . Let $P (2 cos α, 2 sin α), 0 < α < \frac{π}{2}$ and $Q (2 cos β, 2 sin β)$ be two points such that $(α - β) = \frac{π}{2}$ . Then the point of intersection of AQ and BP lies on :
1. A. $x^{2} + y^{2} - 4 x - 4 = 0$
2. B. $x^{2} + y^{2} - 4 x - 4 y = 0$
3. C. $x^{2} + y^{2} - 4 x - 4 y - 4 = 0$
4. D. $x^{2} + y^{2} - 4 y - 4 = 0$
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Answer: D
Q42026 · 24 Jan 2026 · Shift 1
Let a circle of radius 4 pass through the origin O , the points $A (- 3 a, 0)$ and $B (0, - 2 b)$ , where $a$ and $b$ are real parameters and $ab \neq = 0$ . Then the locus of the centroid of $△ OAB$ is a circle of radius
1. A. $\frac{7}{3}$
2. B. $\frac{11}{3}$
3. C. $\frac{5}{3}$
4. D. $\frac{8}{3}$
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Answer: D
Q152026 · 23 Jan 2026 · Shift 2
If the points of intersection of the ellipses $x^{2} + 2 y^{2} - 6 x - 12 y + 23 = 0$ and $4 x^{2} + 2 y^{2} - 20 x - 12 y + 35 = 0$ lie on a circle of radius $r$ and centre $(a, b)$ , then the value of $ab + 18 r^{2}$ is :
1. A.53
2. B.52
3. C.55
4. D.51
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Answer: C
Q182026 · 23 Jan 2026 · Shift 2
The area of the region enclosed between the circles $x^{2} + y^{2} = 4$ and $x^{2} + (y - 2)^{2} = 4$ is:
1. A. $\frac{2}{3} (4 π - 33)$
2. B. $\frac{4}{3} (2 π - 3)$
3. C. $\frac{4}{3} (2 π - 33)$
4. D. $\frac{2}{3} (2 π - 33)$
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Answer: A
Q192026 · 23 Jan 2026 · Shift 2
An equilateral triangle OAB is inscribed in the parabola $y^{2} = 4 x$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having $A B$ as a diameter from the origin is
1. A. $2 (8 - 33)$
2. B. $4 (6 + 3)$
3. C. $4 (3 - 3)$
4. D. $2 (3 + 3)$
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Answer: C
Q62026 · 22 Jan 2026 · Shift 1
Let the set of all values of $r$ , for which the circles $(x + 1)^{2} + (y + 4)^{2} = r^{2}$ and $x^{2} + y^{2} - 4 x - 2 y - 4 = 0$ intersect at two distinct points be the interval $(α, β)$ . Then $α β$ is equal to
1. A.21
2. B.24
3. C.20
4. D.25
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Answer: D
Q112026 · 21 Jan 2026 · Shift 1
Let PQ and MN be two straight lines touching the circle $x^{2} + y^{2} - 4 x - 6 y - 3 = 0$ at the points $A$ and $B$ respectively. Let $O$ be the centre of the circle and $∠ A O B = π /3$ . Then the locus of the point of intersection of the lines PQ and MN is :
1. A. $x^{2} + y^{2} - 18 x - 12 y - 25 = 0$
2. B. $x^{2} + y^{2} - 12 x - 18 y - 25 = 0$
3. C. $3 (x^{2} + y^{2}) - 12 x - 18 y - 25 = 0$
4. D. $3 (x^{2} + y^{2}) - 18 x - 12 y + 25 = 0$
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Answer: C
Q222026 · 21 Jan 2026 · Shift 2· Numerical answer
If $P$ is a point on the circle $x^{2} + y^{2} = 4, Q$ is a point on the straight line $5 x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of PQ , then 13 times the sum of abscissa of all such points P is ____.
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Answer: 2