Circles — JEE Main PYQs

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 $2x + k^2 y - 4 = 0$. If the line $x - y + 2 = 0$ intersects the circle at the points A and B, then $(AB)^2$ is equal to :

Suppose that 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:

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:

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 :

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 $27\sqrt{3}$. Then the square of the length of the chord of the circle on the line $x=1$ is ________.

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 $6\sqrt{3}$ on $y$-axis, then the length of the chord of the circle C on the line $x - y = 3$ is :

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 ________.

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 __________.

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 __________.

If the four distinct points $(4, 6)$, $(-1, 5)$, $(0, 0)$ and $(k, 3k)$ lie on a circle of radius $r$, then $10k + r^2$ is equal to

Let $\mathrm{C}_1$ be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let $\mathrm{C}_2$ be the circle with centre $(1, 3)$ that touches $\mathrm{C}_1$ externally at the point $(\alpha, \beta)$. If $(\beta - \alpha)^2 = \frac{m}{n}$, $\gcd(m, n) = 1$, then $m + n$ is equal to

Let $r$ be the radius of the circle, which touches $x$-axis at point $(a, 0)$, $a < 0$ and the parabola $y^2 = 9x$ at the point $(4, 6)$. Then $r$ is equal to __________

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 $(\alpha, \beta)$, then $3\beta-2\alpha$ is equal to :

Let the parabola $y=x^2+px-3$, 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 $\Delta$PQR is :

The focus of the parabola $y^2 = 4x + 16$ is the centre of the circle C of radius 5. If the values of $\lambda$, for which C passes through the point of intersection of the lines $3x - y = 0$ and $x + \lambda y = 4$, are $\lambda_1$ and $\lambda_2$, $\lambda_1 < \lambda_2$, then $12\lambda_1 + 29\lambda_2$ is equal to _________.

Let circle C be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and A be the point on C such that OA is parallel to x-axis and A lies on the right hand side of the centre O of C. If $B(\alpha, \beta)$, with $\beta < 4$, lies on C such that the length of the arc AB is $(1/6)^{\text{th}}$ of the perimeter of C, then $\beta - \sqrt{3}\alpha$ is equal to

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 $x^2 + y^2 - \alpha x + \beta y + \gamma = 0$. If the circle lies below x-axis, then the ordered pair $(2a, b^2)$ is equal to

A square is inscribed in the circle $x^{2}+y^{2}-10x-6y+30=0$. One side of this square is parallel to $y=x+3$. If $(x_{i}, y_{i})$ are the vertices of the square, then $\Sigma(x_{i}^{2}+y_{i}^{2})$ is equal to :

Let C be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x + y = 2$ intersects 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

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point $(3, 2)$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point $(5, 5)$ is :

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 :

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 :

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m + n^2$ is equal to

Let $C$ be the circle of minimum area touching the parabola $y = 6 - x^2$ and the lines $y = \sqrt{3}\,|x|$. Then, which one of the following points lies on the circle $C$?

If the locus of the point, whose distances from the point $(2, 1)$ and $(1, 3)$ are in the ratio $5:4$, is $\mathrm{a}x^2+\mathrm{b}y^2+\mathrm{c}xy+\mathrm{d}x+\mathrm{e}y+170=0$, then the value of $\mathrm{a}^2+2\mathrm{b}+3\mathrm{c}+4\mathrm{d}+\mathrm{e}$ is equal to :

If $P(6, 1)$ be the orthocentre of the triangle whose vertices are $A(5, -2)$, $B(8, 3)$ and $C(\mathrm{h}, \mathrm{k})$, then the point $C$ lies on the circle :

Let the circles $C_1 : (x-\alpha)^2+(y-\beta)^2=r_1^2$ and $C_2 : (x-8)^2+\left(y-\frac{15}{2}\right)^2=r_2^2$ 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 $(\alpha+\beta)+4\left(r_1^2+r_2^2\right)$ equals

If the image of the point $(-4, 5)$ in the line $x + 2y = 2$ lies on the circle $(x+4)^2 + (y-3)^2 = r^2$, then $r$ is equal to :

Let a circle passing through $(2, 0)$ have its centre at the point $(\mathrm{h}, \mathrm{k})$. Let $(x_{\mathrm{c}}, y_{\mathrm{c}})$ be the point of intersection of the lines $3x+5y=1$ and $(2+\mathrm{c})x+5\mathrm{c}^2y=1$. If $\mathrm{h}=\lim\limits_{\mathrm{c}\to 1} x_{\mathrm{c}}$ and $\mathrm{k}=\lim\limits_{\mathrm{c}\to 1} y_{\mathrm{c}}$, then the equation of the circle is :

Let the centre of a circle, passing through the points $(0, 0)$, $(1, 0)$ and touching the circle $x^2+y^2=9$, be $(\mathrm{h}, \mathrm{k})$. Then for all possible values of the coordinates of the centre $(\mathrm{h}, \mathrm{k})$, $4(\mathrm{h}^2+\mathrm{k}^2)$ is equal to ________.

Consider the circle $C: x^2+y^2=4$ and the parabola $P: y^2=8x$. If the set of all values of $\alpha$, for which three chords of the circle $C$ on three distinct lines passing through the point $(\alpha, 0)$ are bisected by the parabola $P$ is the interval $(p, q)$, then $(2q-p)^2$ is equal to ________.

Let $C: x^2 + y^2 = 4$ and $C': x^2 + y^2 - 4\lambda x + 9 = 0$ be two circles. If the set of all values of $\lambda$ so that the circles $C$ and $C'$ intersect at two distinct points, is $\mathbb{R} - [a, b]$, then the point $(8a + 12, 16b - 20)$ lies on the curve :

Let the line $L: \sqrt{2}x + y = \alpha$ pass through the point of the intersection P (in the first quadrant) of the circle $x^2 + y^2 = 3$ and the parabola $x^2 = 2y$. Let the line L touch two circles $C_1$ and $C_2$ of equal radius $2\sqrt{3}$. 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 $PQ_1Q_2$ is equal to ________.

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 :

Four distinct points $(2k, 3k)$, $(1, 0)$, $(0, 1)$ and $(0, 0)$ lie on a circle for $k$ equal to :

Equations of two diameters of a circle are $2x - 3y = 5$ and $3x - 4y = 7$. The line joining the points $\left(-\dfrac{22}{7}, -4\right)$ and $\left(-\dfrac{1}{7}, 3\right)$ intersects the circle at only one point $P(\alpha, \beta)$. Then, $17\beta - \alpha$ is equal to _______.

If the circles $(x+1)^2+(y+2)^2=r^2$ and $x^2+y^2-4x-4y+4=0$ intersect at exactly two distinct points, then

Consider two circles $C_1: x^2+y^2=25$ and $C_2: (x-\alpha)^2+y^2=16$, where $\alpha \in (5, 9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_1$ and $C_2$ be $\sin^{-1}\left(\frac{\sqrt{63}}{8}\right)$. If the length of common chord of $C_1$ and $C_2$ is $\beta$, then the value of $(\alpha\beta)^2$ equals _______.

If one of the diameters of the circle $x^2+y^2-10x+4y+13=0$ is a chord of another circle C, whose center is the point of intersection of the lines $2x+3y=12$ and $3x-2y=5$, then the radius of the circle C is :

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 $OA+OB$, where $O$ is the origin, is equal to