Circle — JEE Main PYQs

Let A be the point $(1,2)$ and B be any point on the curve $x^2+y^2=16$. If the centre of the locus of the point P , which divides the line segment AB in the ratio $3:2$ is the point $\mathrm{C}(\alpha ,\beta )$, then the length of the line segment AC is

A line segment AB of length $\lambda$ moves such that the points A and B remain on the periphery of a circle of radius $\lambda$. Then the locus of the point, that divides the line segment AB in the ratio $2:3$, is a circle of radius

Two circles in the first quadrant of radii ${\mathrm{r}}_1$ and ${\mathrm{r}}_2$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $x+y=2$. Then $r_1^2+r_2^2-r_1{r}_2$ is equal to ______.

Let the centre of a circle C be $(\alpha ,\beta )$ and its radius $r<8$. Let $3x+4y=24$ and $3x-4y=32$ be two tangents and $4x+3y=1$ be a normal to C. Then $(\alpha -\beta +r)$ is equal to

The number of common tangents, to the circles $x^2+y^2-18x-15y+131=0$ and ${\mathrm{x}}^2+{\mathrm{y}}^2-6\mathrm{x}-6\mathrm{y}-7=0$, is:

Let $y=x+2,4y=3x+6$ and $3y=4x+1$ be three tangent lines to the circle $(x-h{)}^2+(y-k{)}^2=r^2$. Then $h+k$ is equal to:

A triangle is formed by the tangents at the point $(2,2)$ on the curves $y^2=2x$ and $x^2+y^2=4x$, and the line $x+y+2=0$. If r is the radius of its circumcircle, then $r^2$ is equal to

A circle with centre $(2,3)$ and radius 4 intersects the line $x+y=3$ at the points P and Q. If the tangents at P and Q intersect at the point $S(\alpha ,\beta )$, then $4\alpha -7\beta$ is equal to

Let the tangents at the points $\mathrm{A}(4,-11)$ and $\mathrm{B}(8,-5)$ on the circle $x^2+y^2-3x+10y-15=0$, intersect at the point C. Then the radius of the circle, whose centre is C and the line joining A and B is its tangent, is equal to

Points $P(-3,2),Q(9,10)$ and $R(\alpha ,4)$ lie on a circle $C$ with PR as its diameter. The tangents to C at the points Q and R intersect at the point S . If S lies on the line $2\mathrm{x}-\mathrm{ky}=1$, then k is equal to ____ .

The locus of the mid points of the chords of the circle $C_1:(x-4{)}^2+(y-5{)}^2=4$ which subtend an angle ${\theta }_i$ at the centre of the circle $C_1$, is a circle of radius $r_i$. If ${\theta }_1=\frac{\pi }{3},{\theta }_3=\frac{2\pi }{3}$ and …

Let $\mathrm{P}\left({\mathrm{a}}_1,{\mathrm{b}}_1\right)$ and $\mathrm{Q}\left({\mathrm{a}}_2,{\mathrm{b}}_2\right)$ be two distinct points on a circle with center $\mathrm{C}(\sqrt{2},\sqrt{3})$. Let O be the origin and OC be perpendicular to both CP and CQ . If the …

Let a circle ${\mathrm{C}}_1$ be obtained on rolling the circle $x^2+y^2-4x-6y+11=0$ upwards 4 units on the tangent T to it at the point $(3,2)$. Let ${\mathrm{C}}_2$ be the image of ${\mathrm{C}}_1$ in T . Let A and B be the centers of circles ${\mathrm{C}}_1$ …

The points of intersection of the line $\mathrm{ax}+$ by $=0$, $(\mathrm{a}\ne \mathrm{b})$ and the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}=0$ are $A(\alpha ,0)$ and $B(1,\beta )$. The image of the circle with AB as a diameter in the line $\mathrm{x}+\mathrm{y}+2=0$ …

Let AB be a chord of length 12 of the circle $(x-2{)}^2+(y+1{)}^2=\frac{169}{4}$.

If tangents drawn to the circle at points A and B intersect at the point P, then five times the distance of point P from chord AB is equal to ____.

Let C be the centre of the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-\mathrm{x}+2\mathrm{y}=\frac{11}{4}$ and P be a point on the circle. A line passes through the point C , makes an angle of $\frac{\pi }{4}$ with the line CP and intersects the circle at the points Q and R . The …

Let the tangents at two points A and B on the circle $x^2+y^2-4x+3=0$ meet at origin $\mathrm{O}(0,0)$. Then the area of the triangle of OAB is

If the circle $x^2+y^2-2gx+6y-19c=0$, $g,c\in \mathbb{R}$ passes through the point $(6,1)$ and its centre lies on the line $x-2cy=8$, then the length of intercept made by the circle on x-axis is

Let the abscissae of the two points P and Q on a circle be the roots of $x^2-4x-6=0$ and the ordinates of P and Q be the roots of $y^2+2y-7=0$. If PQ is a diameter of the circle $x^2+y^2+2ax+2by+c=0$, then the value of $(a+b-c)$ is

A circle ${\mathrm{C}}_1$ passes through the origin O and has diameter 4 on the positive x -axis. The line $\mathrm{y}=2\mathrm{x}$ gives a chord OA of a circle ${\mathrm{C}}_1$. Let ${\mathrm{C}}_2$ be the circle with OA as a diameter. If the tangent to …

If the circles $x^2+y^2+6x+8y+16=0$ and $x^2+y^2+2(3-\sqrt{3})x+2(4-\sqrt{6})y=\mathrm{k}+6\sqrt{3}+8\sqrt{6},\mathrm{k}>0$, touch internally at the point $P(\alpha ,\beta )$, then $(\alpha +\sqrt{3}{)}^2+(\beta +\sqrt{6}{)}^2$ is equal to

A point P moves so that the sum of squares of its distances from the points $(1,2)$ and $(-2,1)$ is 14 . Let $\mathrm{f}(\mathrm{x},\mathrm{y})=0$ be the locus of P , which intersects the x -axis at the points $\mathrm{A},\mathrm{B}$ and the y -axis at the point …

If one of the diameters of the circle $x^2+y^2-2\sqrt{2}x-6\sqrt{2}y+14=0$ is a chord of the circle $(x-2\sqrt{2}{)}^2+(y-2\sqrt{2}{)}^2=r^2$, then the value of $r^2$ is equal to

The set of values of $k$ for which the circle $C:4x^2+4y^2-12x+8y+k=0$ lies inside the fourth quadrant and the point $\left(1,-\frac{1}{3}\right)$ lies on or inside the circle $C$ is :

Let a circle C of radius 5 lie below the x-axis. The line $L_1:4x+3y+2=0$ passes through the centre $P$ of the circle C and intersects the line ${\mathrm{L}}_2:3\mathrm{x}-4\mathrm{y}-11=0$ at Q. The line ${\mathrm{L}}_2$ touches C at the point Q. Then the distance of P fro …

If the tangents drawn at the point $\mathrm{O}(0,0)$ and $P(1+\sqrt{5},2)$ on the circle $x^2+y^2-2x-4y=0$ intersect at the point Q , then the area of the triangle OPQ is equal to

Let the lines $y+2x=\sqrt{11}+7\sqrt{7}$ and $2y+x=2\sqrt{11}+6\sqrt{7}$ be normal to a circle $C:(x-h{)}^2+(y-k{)}^2=r^2$. If the line $\sqrt{11}y-3x=\frac{5\sqrt{77}}{3}+11$ is tangent to the circle C, then the value of $(5h-8k{)}^2+5r^2$ is equal to ____.

Let the tangent to the circle ${\mathrm{C}}_1:x^2+y^2=2$ at the point $\mathrm{M}(-1,1)$ intersect the circle ${\mathrm{C}}_2$: $(x-3{)}^2+(y-2{)}^2=5$, at two distinct points A and B . If the tangents to ${\mathrm{C}}_2$ at the points A and B intersect at N , then the area …

Let C be a circle passing through the points $\mathrm{A}(2,-1)$ and $\mathrm{B}(3,4)$. The line segment AB is not a diameter of C . If r is the radius of C and its centre lies on the circle $(x-5{)}^2+(y-1{)}^2=\frac{13}{2}$, then $r^2$ is equal to :

A rectangle R with end points of one of its sides as $(1,2)$ and $(3,6)$ is inscribed in a circle. If the equation of a diameter of the circle is $2x-y+4=0$, then the area of R is ______.

Let the mirror image of a circle $c_1:x^2+y^2-2x-6y+\alpha =0$ in line $y=x+1$ be $c_2:5x^2+5y^2+10gx+10\mathrm{fy}+38=0$. If r is the radius of circle ${\mathrm{c}}_2$, then $\alpha +6{\mathrm{r}}^2$ is equal to ______.

Let $x^2+y^2+Ax+By+C=0$ be a circle passing through $(0,6)$ and touching the parabola $\mathrm{y}={\mathrm{x}}^2$ at (2, 4). Then $\mathrm{A}+\mathrm{C}$ is equal to

Let a circle $C:(x-h{)}^2+(y-k{)}^2=r^2,k>0$, touch the x-axis at $(1,0)$. If the line $x+y=0$ intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of $h+k+r$ is equal to ____.

A circle touches both the y -axis and the line $x+y=0$. Then the locus of its center is

Let a circle C touch the lines ${\mathrm{L}}_1:4\mathrm{x}-3\mathrm{y}+{\mathrm{K}}_1=0$ and ${\mathrm{L}}_2:4\mathrm{x}-3\mathrm{y}+{\mathrm{K}}_2=0,{\mathrm{K}}_1,{\mathrm{K}}_2\in \mathrm{R}$. If a line passing through the centre of the circle C intersects …

Let the abscissae of the two points P and Q be the roots of $2{\mathrm{x}}^2-\mathrm{rx}+\mathrm{p}=0$ and the ordinates of P and Q be the roots of ${\mathrm{x}}^2-\mathrm{sx}-\mathrm{q}=0$. If the equation of the circle described on PQ as diameter is …

If a line along a chord of the circle $4x^2+4y^2+120x+675=0$, passes through the point $(-30,0)$ and is tangent to the parabola $y^2=30x$, then the length of this chord is:

The locus of a point which moves such that the sum of squares of its distances from the points $(0,0),(1,0),(0,1),(1,1)$ is 18 units, is a circle of diameter $d$. Then $d^2$ is equal to ____.

If one of the diameters of the circle $x^2+y^2-2x-6y+6=0$ is a chord of another circle '$C$', whose centre is at $(2,1)$, then its radius is ______.

A circle $C$ touches the line $x=2y$ at the point $(2,1)$ and intersects the circle $C_1:x^2+y^2+2y-5=0$ at two points $P$ and $Q$ such that $PQ$ is a diameter of $C_1$. Then the diameter of $C$ is

If the area of the triangle formed by the positive $x$-axis, the normal and the tangent to the circle $(x-2{)}^2+(y-3{)}^2=25$ at the point $(5,7)$ is $A$, then $24A$ is equal to ____

Let $r_1$ and $r_2$ be the radii of the largest and smallest circles, respectively, which pass through the point $(-4,1)$ and having their centres on the circumference of the circle $x^2+y^2+2x+4y-4=0$. If $\frac{r_1}{r_2}=a+b\sqrt{2}$, then $a+b$ is eq …

Two circles each of the radius 5 units touch each other at the point ( 1,2 ). If the equation of their common tangent is $4x+3y=10$, and $C_1(\alpha ,\beta )$ and $C_2(\gamma ,\delta ),C_1\ne C_2$ are their centres, then $|(\alpha +\beta )(\gamma +\delta )|$ is equal to __ …

The locus of the point of intersection of the lines $(\sqrt{3})kx+ky-4\sqrt{3}=0$ and $\sqrt{3}x-y-4(\sqrt{3})k=0$ is a conic, whose eccentricity is ____

Let the circle $S:36x^2+36y^2-108x+120y+C=0$ be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, $x-2y=4$ and $2x-y=5$ lies inside the circle S, then :

In the circle given below, let $\mathrm{OA}=1$ unit, $\mathrm{OB}=13$ unit and $\mathrm{PQ}\perp \mathrm{OB}$. Then, the area of the triangle PQB (in square units) is :

Let the equation $x^2+y^2+px+(1-p)y+5=0$ represent circles of varying radius $r\in (0,5]$. Then the number of elements in the set $S=\{q:q=p^2$ and $q$ is an integer $\}$ is

If the locus of the mid-point of the line segment from the point $(3,2)$ to a point on the circle, $x^2+y^2=1$ is a circle of radius $r$, then $r$ is equal to :

Let $B$ be the centre of the circle $x^2+y^2-2x+4y+1=0$. Let the tangents at two points $P$ and $Q$ on the circle intersect at the point $A(3,1)$. Then $8\cdot \left(\frac{\text{ area }△APQ}{\text{ area }△BPQ}\right)$ is equal to ____.

If the variable line lies between the two circles and , without intercepting a chord on either circle, then the sum of all the integral values of is ______.