Circles — JEE Maths Practice Questions & Solutions

17 questions on Circles with full step-by-step solutions, including past-year (PYQ) problems. Free to practice.

hardPYQ · JEE Main 2026

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

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mediumPYQ · JEE Main 2026

Let a circle pass through the origin and its centre be the point of intersection of the 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

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mediumPYQ · JEE Main 2026

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 $72e^2$ is equal to

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mediumPYQ · JEE Main 2026

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

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mediumPYQ · JEE Main 2026

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

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hard

Each entry of Column-I is to be matched with its value in Column-II.

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medium

A ray of light strikes the point $(-3,-2)$ on the tangent drawn at $(0,-2)$ to the circle $x^2+y^2=4$, and after reflection it touches that circle. Given that the incident ray moved along the line $Jx-Ky+46=0$, find the value of $J+K$.

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medium

A ray starting from the point $A(\cos\theta,\sin\theta)$ strikes a mirror at a variable point $P$ and, after reflection, passes through the point $B(-\sin\theta,\cos\theta)$, the incident ray and the reflected ray both being tangents to the circle $x^2+y^2=1$. The variable point $P$ then traces $x^2+y^2=k$. Find the value of $k$.

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medium

Let $C_1$ and $C_2$ be two concentric circles of radii $r_1$ and $r_2$ $(r_1<r_2)$. Suppose that from every point of $C_2$ the two tangents drawn to $C_1$ meet $C_2$ again at the ends of a diameter of $C_2$. Then

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medium

Let the two circles be $S_1 \equiv x^2+y^2-4x-6y-12=0$ and $S_2 \equiv x^2+y^2+6x+4y-12=0$, and let the line be $L \equiv x+y=0$. Then

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medium

From the point $(6,4)$ two tangents are drawn to the circle $x^2+y^2-4x-4y+4=0$, and they cut the $y$-axis at $A$ and $B$. The length $AB$ equals

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medium

For a circle, the line $y=x-2$ is a normal, the abscissa of the centre is twice its ordinate, and the circle touches the $x$-axis. The equation of the circle is

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medium

In the figure alongside, a chord $AB$ of a circle cuts a diameter at right angles at the point $T$, and the two pieces of that diameter, of lengths $2x$ and $2y$, are the diameters of the two smaller circles drawn inside. The chord $AB$ has length $4$. Find the area of the shaded region, that is, the big circle with the two smaller circles removed.

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medium

Let $C_1$ and $C_2$ be two concentric circles centred at the origin, of radii $4$ and $2$ units. Shade the part of the annulus between them that lies in the first quadrant above the line $y=x$, that is, the annular sector from $45^\circ$ to $90^\circ$. If the point $(a,3)$ lies inside this shaded region, the interval of $a$ is

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medium

Given that a circle touching $(x-2)^2+y^2=1$ and also the angle bisectors of the transverse common tangents of $(x\pm2)^2+y^2=1$ has radius $p\pm\sqrt q$, the value of $p+q$ is

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medium

Given the circles $x^2+y^2-4x-4y+4=0$ and $x^2+y^2+6x+6y+9=0$, their internal common tangents are

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medium

Let $AB$ be a chord of the circle $x^2+y^2-6x-4y+3=0$ whose mid point is $(1,1)$. The tangents drawn at $A$ and $B$ meet at the point

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