Circle — JEE Main PYQs

Let
$A=\left\{(x,y)\in R\times R\mid 2x^2+2y^2-2x-2y=1\right\}$,
$B=\left\{(x,y)\in R\times R\mid 4x^2+4y^2-16y+7=0\right\}$ and
$C=\left\{(x,y)\in R\times R\mid x^2+y^2-4x-2y+5\le r^2\right\}$.
Then the minimum value of $|r|$ such that …

Two tangents are drawn from the point $P(-1,1)$ to the circle $x^2+y^2-2x-6y+6=0$. If these tangents touch the circle at points A and B , and if D is a point on the circle such that length of the segments $AB$ and $AD$ are equal, then the area of the triangle …

Let P and Q be two distinct points on a circle which has center at $\mathrm{C}(2,3)$ and which passes through origin O . If OC is perpendicular to both the line segments CP and CQ , then the set $\{\mathrm{P},\mathrm{Q}\}$ is equal to :

Choose the incorrect statement about the two circles whose equations are given below:
$x^2+y^2-10x-10y+41=0$ and $x^2+y^2-16x-10y+80=0$

The line $2x-y+1=0$ is a tangent to the circle at the point $(2,5)$ and the centre of the circle lies on $x-2y=4$. Then, the radius of the circle is :

The minimum distance between any two points $P_1$ and $P_2$ while considering point $P_1$ on one circle and point $P_2$ on the other circle for the given circles' equations
$x^2+y^2-10x-10y+41=0$
$x^2+y^2-24x-10y+160=0$ is ____

The equation of a circle is $\mathrm{Re}\left(z^2\right)+2(\mathrm{Im}(z){)}^2+2\mathrm{Re}(z)=0$, where $z=x+iy$. A line which passes through the center of the given circle and the vertex of the parabola, $x^2-6x-y+13=0$, has $y$ intercept equal …

Consider a circle $C$ which touches the $y$-axis at $(0,6)$ and cuts off an intercept $6\sqrt{5}$ on the $x$-axis. Then the radius of the circle $C$ is equal to :

Let ABCD be a square of side of unit length. Let a circle ${\mathrm{C}}_1$ centered at A with unit radius is drawn. Another circle $C_2$ which touches $C_1$ and the lines $AD$ and $AB$ are tangent to it, is also drawn. Let a tangent line from the point C to the circle …

Let the lengths of intercepts on $x$-axis and $y$-axis made by the circle $x^2+y^2+ax+2ay+c=0,(a<0)$ be $2\sqrt{2}$ and $2\sqrt{5}$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line $x+2y=0$, is …

Let $S_1:x^2+y^2=9$ and $S_2:(x-2{)}^2+y^2=1$. Then the locus of center of a variable circle $S$ which touches $S_1$ internally and $S_2$ externally always passes through the points :

Two tangents are drawn from a point $P$ to the circle $x^2+y^2-2x-4y+4=0$, such that the angle between these tangents is ${\tan }^{-1}\left(\frac{12}{5}\right)$, where ${\tan }^{-1}\left(\frac{12}{5}\right)\in (0,\pi )$. If the centre of the circle is denoted by $C$ a …

Let the tangent to the circle $x^2+y^2=25$ at the point $R(3,4)$ meet $x$-axis and $y$-axis at points $P$ and $Q$, respectively. If $r$ is the radius of the circle passing through the origin $O$ and having centre at the incentre of the triangle OPQ , then …

For the four circles $\mathrm{M},\mathrm{N},\mathrm{O}$ and P , following four equations are given :
Circle $\mathrm{M}:{\mathrm{x}}^2+{\mathrm{y}}^2=1$
Circle $\mathrm{N}:{\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}=0$
Circle …

Choose the correct statement about two circles whose equations are given below:

Let the tangents drawn from the origin to the circle, $x^2+y^2-8x-4y+16=0$ touch it at the points $A$ and $B$. Then $(AB{)}^2$ is equal to:

If a line, $y=mx+c$ is a tangent to the circle, $(x-3{)}^2+y^2=1$ and it is perpendicular to a line $L_1$, where $L_1$ is the tangent to the circle, $x^2+y^2=1$ at the point $\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$; then:

If the curves, $x^2-6x+y^2+8=0$ and $x^2-8y+y^2+16-k=0,(k>0)$ touch each other at a point, then the largest value of $k$ is ______.

A circle touches the $y$-axis at the point $(0,4)$ and passes through the point $(2,0)$. Which of the following lines is not a tangent to this circle?

If the length of the chord of the circle, $x^2+y^2=r^2(r>0)$ along the line $y-2x=3$ is $r$, the $r^2$ is equal to:

The diameter of the circle, whose centre lies on the lines $x+y=2$ in the first quadrant and which touches both the lines $x=3$ and $y=2$, is ______.

The centre of the circle passing through the point $(0,1)$ and touching the parabola $y=x^2$ at the point $(2,4)$ is:

The circle passing through the intersection of the circles, $x^2+y^2-6x=0$ and $x^2+y^2-4y=0$, having its centre on the line, $2x-3y+12=0$, also passes through the point:

Let PQ be a diameter of the circle $x^2+y^2=9$. If $\alpha$ and $\beta$ are the lengths of the perpendiculars from P and Q on the straight line, $\mathrm{x}+\mathrm{y}=2$ respectively, then the maximum value of $\alpha \beta$ is

The sum of the squares of the lengths of the chords intercepted on the circle, $x^2+y^2=16$, by the lines, $x+y=n,n\in N$, where $N$ is the set of all natural numbers is:

The tangent and the normal lines at the point $(\sqrt{3},1)$ to the circle $x^2+y^2=4$ and the $x$-axis form a triangle. The area of this triangle (in square units) is:

If a tangent to the circle $x^2+y^2=1$ intersects the coordinate axes at distinct points $P$ and $Q$, then the locus of the mid-point of $PQ$ is:

If a circle $C$ passing through the point $(4,0)$ touches the circle $x^2+y^2+4x-6y=12$ externally at the point $(1,-1)$, then the radius of $C$ is:

A circle cuts a chord of length 4a on the $x$-axis and passes through a point on the $y$-axis, distant 2b from the origin. Then the locus of the center of this circle, is:

If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is $90^{\circ }$, then the length (in cm ) of their common chord is :

If the line $ax+y=c$, touches both the curves $x^2+y^2=1$ and $y^2-4\sqrt{2}x$, then $|c|$ is equal to

The locus of the centres of the circles, which touch the circle, $x^2+y^2=1$ externally, also touch the $y$-axis and lie in the first quadrant is

A square is inscribed in the circle $x^2+y^2-6x+8y-103=0$ with its sides parallel to the coordinate axes. Then the distance of the vertex of the square which is nearest to the origin is:

Two circles with equal radii intersecting at the points $(0,1)$ and $(0,-1)$. The tangent at the point $(0,1)$ to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is:

The straight line $x+2y=1$ meets the coordinate axes at A and B. A circle is drawn through $\mathrm{A},\mathrm{B}$ and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is:

Let ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ be the centres of the circles ${\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}-2\mathrm{y}-2=0$ and ${\mathrm{x}}^2+{\mathrm{y}}^2-6\mathrm{x}-6y+14=0$ respectively. If P and Q are the points of intersection of these circles, then the area …

If a variable line, $3x+4y-\lambda =0$ is such that the two circles $x^2+y^2-2x-2y+1=0$ and $x^2+y^2-18x-2y+78=0$ are on its opposite sides, then the set of all values of $\lambda$ is the interval:

A circle touching the $x$-axis at $(3,0)$ and making an intercept of length 8 on the $y$-axis passes through the point :

If a circle of radius R passes through the origin O and intersects the coordinate axes at A and $B$, then the locus of the foot of perpendicular from $O$ on $AB$ is :

The common tangent to the circles $x^2+y^2=4$ and $x^2+y^2+6x+8y-24=0$ also passes through the point

A rectangle is inscribed in a circle with a diameter lying along the line $3y=x+7$. If the two adjacent vertices of the rectangle are $(-8,5)$ and $(6,5)$ then the area of the rectangle (in sq. units) is

If the circles $x^2+y^2-16x-20y+164=r^2$ and $(x-4{)}^2+(y-7{)}^2=36$ intersect at two distinct points, then:

Three circles of radii $\mathrm{a},\mathrm{b},\mathrm{c}$ $(\mathrm{a}<\mathrm{b}<\mathrm{c})$ touch each other externally. If they have x-axis as a common tangent, then:

The line $x=y$ touches a circle at the point $(1,1)$. If the circle also passes through the point $(1,-3)$, then its radius is

If the circles $x^2+y^2+5Kx+2y+K=0$ and $2\left(x^2+y^2\right)+2Kx+3y-1=0,(K\in R)$, intersect at the points $P$ and $Q$, then the line $4x+5y-K=0$ passes through $P$ and $Q$ for:

If the tangent at $(1,7)$ to the curve $x^2=y-6$ touches the circle $x^2+y^2+16x+12y+\mathrm{c}=0$ then the value of c is :

The centres of those circles which touch the circle, $x^2+y^2-8x-8y-4=0$, externally and also touch the x-axis, lie on:

If one of the diameters of the circle, given by the equation, $x^2+y^2-4x+6y-12=0$, is a chord of a circle S, whose centre is at $(-3,2)$, then the radius of S is:

The number of common tangents to the circles $x^2+y^2-4x-6y-12=0$ and $x^2+y^2+6x+18y+26=0$, is :

Let C be the circle with centre at $(1,1)$ and radius $=1$. If T is the circle centred at $(0,y)$, passing through origin and touching the circle C externally, then the radius of T is equal to