Circles

14 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 1·Q15·MCQ·Moderate
Open
Locus from tangents to circle
Let $PQ$ and $MN$ be two straight lines touching the circle $x^2 + y^2 - 4x - 6y - 3 = 0$ at the points $A$ and $B$ respectively. Let $O$ be the centre of the circle and $\angle AOB = \pi/3$ . Then the locus of the point of intersection of the lines $PQ$ and $MN$ is:
1. (1) $3(x^2 + y^2) - 18x - 12y + 25 = 0$
2. (2) $x^2 + y^2 - 12x - 18y - 25 = 0$
3. (3) $x^2 + y^2 - 18x - 12y - 25 = 0$
4. (4) $3(x^2 + y^2) - 12x - 18y - 25 = 0$
MathsJEE Main 22 Jan 2026, Shift 1·Q12·MCQ·Easy–moderate
Open
Intersecting circles condition
Let the set of all values of $r$ , for which the circles $(x + 1)^2 + (y + 4)^2 = r^2$ and $x^2 + y^2 - 4x - 2y - 4 = 0$ intersect at two distinct points be the interval $(\alpha, \beta)$ . Then $\alpha\beta$ is equal to
1. (1)25
2. (2)20
3. (3)21
4. (4)24
MathsJEE Main 24 Jan 2026, Shift 1·Q2·MCQ·Easy–moderate
Open
Locus of centroid
Let a circle of radius 4 pass through the origin $O$ , the points $A\left(-\sqrt3\,a, 0\right)$ and $B\left(0, -\sqrt2\,b\right)$ , where $a$ and $b$ are real parameters and $ab \neq 0$ . Then the locus of the centroid of $\Delta OAB$ is a circle of radius
1. (1) $\dfrac53$
2. (2) $\dfrac73$
3. (3) $\dfrac83$
4. (4) $\dfrac{11}{3}$
MathsJEE Main 28 Jan 2026, Shift 1·Q3·MCQ·Moderate
Open
Chord farthest from centre
Let $y = x$ be the equation of a chord of the circle $C_1$ (in the closed half-plane $x \ge 0$ ) 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 $x + ay + b = 0$ , then $a - b$ is equal to:
1. (1)10
2. (2) $-6$
3. (3) $-2$
4. (4)6
MathsJEE Main 28 Jan 2026, Shift 2·Q12·MCQ·Tough
Open
Locus of intersection of chords
Let the circle $x^2 + y^2 = 4$ intersect $x$ -axis at the points $A(a, 0)$ , $a > 0$ and $B(b, 0)$ . Let $P(2\cos\alpha, 2\sin\alpha)$ , $0 < \alpha < \dfrac{\pi}{2}$ and $Q(2\cos\beta, 2\sin\beta)$ be two points such that $(\alpha - \beta) = \dfrac{\pi}{2}$ . Then the point of intersection of $AQ$ and $BP$ lies on :
1. (1) $x^2 + y^2 - 4y - 4 = 0$
2. (2) $x^2 + y^2 - 4x - 4 = 0$
3. (3) $x^2 + y^2 - 4x - 4y = 0$
4. (4) $x^2 + y^2 - 4x - 4y - 4 = 0$
MathsJEE Main 2 Apr 2026, Shift 1·Q24·Numerical·Moderate
Open
Intercepts and chord length
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 $\sqrt{14}$ , then the square of the radius of $C$ is .
MathsJEE Main 2 Apr 2026, Shift 2·Q10·MCQ·Moderate
Open
Length of chord of circle
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 $2x + k^2y - 4 = 0$ . If the line $x - y + 2 = 0$ intersects the circle at the points $A$ and $B$ , then $(AB)^2$ is equal to:
1. (1)10
2. (2)27
3. (3)18
4. (4)34
MathsJEE Main 4 Apr 2026, Shift 1·Q13·MCQ·Moderate
Open
Chords bisected by a line
Suppose the two chords, drawn from the point $(1, 2)$ on the circle $x^2 + y^2 + x - 3y = 0$ 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 $(\alpha, \beta)$ , then $6(\alpha + \beta)$ is equal to:
1. (1)1
2. (2)3
3. (3)4
4. (4)6
MathsJEE Main 5 Apr 2026, Shift 1·Q10·MCQ·Easy–moderate
Open
Maximum distance from a circle
Let $P$ be a moving point on the circle $x^2 + y^2 - 6x - 8y + 21 = 0$ Then, the maximum distance of $P$ from the vertex of the parabola $x^2 + 6x + y + 13 = 0$ is equal to:
1. (1)8
2. (2)10
3. (3)12
4. (4)9
MathsJEE Main 5 Apr 2026, Shift 2·Q11·MCQ·Moderate
Open
Secant through an external point
Let the point $P$ be the vertex of the parabola $y = x^2 - 6x + 12$ . If a line passing through the point $P$ intersects the circle $x^2 + y^2 - 2x - 4y + 3 = 0$ at the points $R$ and $S$ , then the maximum value of $(PR + PS)^2$ is:
1. (1)10
2. (2)20
3. (3)25
4. (4)5
MathsJEE Main 6 Apr 2026, Shift 1·Q22·Numerical·Moderate
Open
Length of chord of a circle
Let the centre of the circle $x^2 + y^2 + 2gx + 2fy + 25 = 0$ be in the first quadrant and lie on the line $2x - y = 4$ . Let the area of an equilateral triangle inscribed in the circle be $27\sqrt3$ . Then the square of the length of the chord of the circle on the line $x = 1$ is .
MathsJEE Main 6 Apr 2026, Shift 2·Q10·MCQ·Easy–moderate
Open
Length of chord of circle
Let $C$ be a circle having centre in the first quadrant and touching the $x$ -axis at a distance 3 units from the origin. If the circle $C$ has an intercept of length $6\sqrt3$ on $y$ -axis, then the length of the chord of the circle $C$ on the line $x - y = 3$ is:
1. (1)8
2. (2)6
3. (3) $6\sqrt2$
4. (4) $8\sqrt2$
MathsJEE Main 6 Apr 2026, Shift 2·Q23·Numerical·Moderate
Open
Points on circle equidistant
Let the line $x - y = 4$ intersect the circle $C : (x - 4)^2 + (y + 3)^2 = 9$ at the points $Q$ and $R$ . If $P(\alpha, \beta)$ is a point on $C$ such that $PQ = PR$ , then $(6\alpha + 8\beta)^2$ is equal to .
MathsJEE Main 8 Apr 2026, Shift 2·Q24·Numerical·Tough
Open
Locus by homogenisation of chord
Consider the circle $C : x^2 + y^2 - 6x - 8y - 11 = 0$ . 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 $x^2 + y^2 - \alpha x - \beta y - \gamma = 0$ , then $\alpha + \beta + 2\gamma$ is equal to .