Circle — JEE Main PYQs

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 $\angle \mathrm{AOB}=\pi /3$. Then the locus of the point of intersection of the lines PQ and MN is:

If P is a point on the circle $x^2+y^2=4$, Q is a point on the straight line $5x+y+2=0$ and $x-y+1=0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such point P is ______.

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

Let a circle of radius 4 pass through the origin O , the points $\mathrm{A}(-\sqrt{3}\mathrm{a},0)$ and $\mathrm{B}(0,-\sqrt{2}\mathrm{b})$, where a and b are real parameters and $\mathrm{ab}\ne 0$. Then the locus of the centroid of $△\mathrm{OAB}$ is a cir …

Let $y=x$ be the equation of a chord of the circle ${\mathrm{C}}_1$ (in the closed half-plane $\mathrm{x}\ge 0$) of diameter 10 passing through the origin. Let ${\mathrm{C}}_2$ be another circle described on the given chord as its diameter. If the equation of the chord …

Let the circle $x^2+y^2=4$ intersect $x$-axis at the points $\mathrm{A}(\mathrm{a},0),\mathrm{a}>0$ and $\mathrm{B}(\mathrm{b},0)$. Let $\mathrm{P}(2\cos \alpha ,2\sin \alpha ),0<\alpha <\frac{\pi }{2}$ and $\mathrm{Q}(2\cos \beta ,2\sin \beta )$ be two points …

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 ______.

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 :

Let P be a moving point on the circle


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 $(\mathrm{PR}+\mathrm{PS}{)}^2$ is:

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

Let the line $\mathrm{x}-\mathrm{y}=4$ intersect the circle
$\mathrm{C}:(\mathrm{x}-4{)}^2+(\mathrm{y}+3{)}^2=9$ at the points Q and R . If $\mathrm{P}(\alpha ,\beta )$ is a point on C such that $\mathrm{PQ}=\mathrm{PR}$, then $(6\alpha +8\beta {)}^2$ is equal to

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 …

Consider the circle $\mathrm{C}:{\mathrm{x}}^2+{\mathrm{y}}^2-6\mathrm{x}-8\mathrm{y}-11=0$.

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 …

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines $\mathrm{x}+(\mathrm{k}-1)\mathrm{y}+3=0$ and $2\mathrm{x}+{\mathrm{k}}^2\mathrm{y}-4=0$. If the line $\mathrm{x}-\mathrm{y}+2=0$ intersects the circle a …

Let the parabola $y=x^2+px-3$, meet the coordinate axes at the points $\mathrm{P},\mathrm{Q}$ and R. If the circle C with centre at $(-1,-1)$ passes through the points $P,Q$ and $R$, then the area of $△PQR$ is :

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 …

Let the circle C touch the line $\mathrm{x}-\mathrm{y}+1=0$, have the centre on the positive x -axis, and cut off a chord of length $\frac{4}{\sqrt{13}}$ along the line $-3x+2y=1$. Let H be the hyperbola $\frac{{\mathrm{x}}^2}{{\alpha }^2}-\frac{{\mathrm{y}}^2}{{\beta }^2}=1$, wh …

Let a circle C pass through the points $(4,2)$ and $(0,2)$, and its centre lie on $3x+2y+2=0$. Then the length of the chord, of the circle C , whose midpoint is $(1,2)$, is:

Let the line $x+y=1$ meet the circle $x^2+y^2=4$ at the points A and B . If the line perpendicular to AB and passing through the mid point of the chord AB intersects is the circle at C and D , then the area of the quadrilateral ADBC is equal to

Let the equation of the circle, which touches x -axis at the point ( $\mathrm{a},0$ ), $\mathrm{a}>0$ 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 …

Let circle $C$ be the image of $x^2+y^2-2x+4y-4=0$ in the line $2\mathrm{x}-3\mathrm{y}+5=0$ 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 $\mathrm{B}(\alpha ,\beta )$, with $\beta <4$, lies on C such …

If the four distinct points $(4,6),(-1,5),(0,0)$ and $(\mathrm{k},3\mathrm{k})$ lie on a circle of radius r , then $10\mathrm{k}+{\mathrm{r}}^2$ 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 ______.

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 …

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 …

Let $r$ be the radius of the circle, which touches x-axis at point ($\mathrm{a},0$), $\mathrm{a}<0$ and the parabola ${\mathrm{y}}^2=9\mathrm{x}$ at the point $(4,6)$. Then r 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 $\sum (x_i^2+y_i^2)$ is equal to

Let $C$ be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x+y=2$ intersect the circle $C$ at the points $P$ and $Q$. Let $MN$ be a chord of $C$ of length 2 units and slope $-1$. Then, the distance (in units) between the chord …

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 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 …

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

If the locus of the point, whose distance from the point $(2,1)$ and $(1,3)$ are in the ratio $5:4$, is $a{x}^2+b{y}^2+cxy+dx+ey+170=0$, then the value of $a^2+2b+3c+4d+e$ is

Let the circles $C_1:(x-\alpha {)}^2+(y-\beta {)}^2=r_1^2$ and ${\mathrm{C}}_2:(\mathrm{x}-8{)}^2+{\left(\mathrm{y}-\frac{15}{2}\right)}^2={\mathrm{r}}_2^2$ touch each other externally at the point (6, 6). If the point (6, 6) divides the line segment joining the centres …

Let a circle passing through $(2,0)$ have its centre at the point ( $\mathrm{h},\mathrm{k}$ ). Let ( ${\mathrm{x}}_{\mathrm{c}},{\mathrm{y}}_{\mathrm{c}}$ ) be the point of intersection of the lines $3x+5y=1$ and $(2+c)x+5c^{2}y=1$. If …

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

Consider the circle $\mathrm{C}:{\mathrm{x}}^2+{\mathrm{y}}^2=4$ 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 …

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 a circle $(x-\alpha {)}^2+(y-\beta {)}^2=50$, where $\alpha ,\beta >0$. If the circle touches the line $y+x=0$ at the point P, whose distance from the origin is $4\sqrt{2}$, then $(\alpha +\beta {)}^2$ is equal to

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

Let the locus of the mid points of the chords of circle $x^2+(y-1{)}^2=1$ drawn from the origin intersect the line $\mathrm{x}+\mathrm{y}=1$ at P and Q . Then, the length of PQ is :

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 $2\mathrm{x}+3\mathrm{y}=12$ and $3\mathrm{x}-2\mathrm{y}=5$, then the radius of the circle C is

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

Let the line $\mathrm{L}:\sqrt{2}\mathrm{x}+\mathrm{y}=\alpha$ 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 ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ of eq …

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

Consider two circles ${\mathrm{C}}_1:{\mathrm{x}}^2+{\mathrm{y}}^2=25$ and ${\mathrm{C}}_2:(\mathrm{x}-\alpha {)}^2+y^2=16$, where $\alpha \in (5,9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of ${\mathrm{C}}_1$ …

If the tangents at the points P and Q on the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}+\mathrm{y}=5$ meet at the point $\mathrm{R}\left(\frac{9}{4},2\right)$, then the area of the triangle PQR is

Consider a circle $C_1:x^2+y^2-4x-2y=\alpha -5$.

Let its mirror image in the line $y=2x+1$ be another circle $C_2:5x^2+5y^2-10fx-10gy+36=0$. Let $r$ be the radius of $C_2$. Then $\alpha +r$ is equal to ______.

A circle passing through the point $\mathrm{P}(\alpha ,\beta )$ in the first quadrant touches the two coordinate axes at the points A and B . The point P is above the line AB . The point Q on the line segment AB is the foot of perpendicular from P on AB . If PQ is equal to 11 un …

Let O be the origin and OP and OQ be the tangents to the circle $x^2+y^2-6x+4y+8=0$ at the point P and Q on it. If the circumcircle of the triangle OPQ passes through the point $\left(\alpha ,\frac{1}{2}\right)$, then a value of $\alpha$ is