Circle — JEE Advanced PYQs

Match each entry in List-I to the correct entry in List-II and choose the correct option.
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Let the straight line $\mathrm{y}=2\mathrm{x}$ touch a circle with center $(0,\alpha ),\alpha >0$, and radius r at a point ${\mathrm{A}}_1$. Let ${\mathrm{B}}_1$ be the point on the circle such that the line segment ${\mathrm{A}}_1{\mathrm{B}}_1$ is a diameter of the …

Let ${\mathrm{C}}_1$ be the circle of radius 1 with center at the origin. Let ${\mathrm{C}}_2$ be the circle of radius r with center at the point $\mathrm{A}=(4,1)$, where $1<\mathrm{r}<3$. Two distinct common tangents PQ and ST of ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ are drawn. …

Let $ABC$ be the triangle with $AB=1,AC=3$ and $\angle BAC=\frac{\pi }{2}$. If a circle of radius $r>0$ touches the sides $AB,AC$ and also touches internally the circumcircle of the triangle $ABC$, then the value of $r$ is ____.

Let G be a circle of radius $R>0$. Let $G_1,G_2,\dots ,G_n$ be n circles of equal radius $r>0$. Suppose each of the n circles $G_1,G_2,\dots ,G_n$ touches the circle G externally. Also, for $i=1,2,\dots ,n-1$, the circle $G_i$ touches $G_{i+1}$ extern …

Consider the region $\mathrm{R}=\{(\mathrm{x},\mathrm{y})\in \mathbb{R}\times \mathbb{R}:\mathrm{x}\ge 0$ and ${\mathrm{y}}^2\le 4-\mathrm{x}\}$. Let $F$ be the family of all circles that are contained in R and have centers on the x-axis. Let C be …

Consider the region $\mathrm{R}=\{(\mathrm{x},\mathrm{y})\in \mathbb{R}\times \mathbb{R}:\mathrm{x}\ge 0$ and ${\mathrm{y}}^2\le 4-\mathrm{x}\}$. Let $F$ be the family of all circles that are contained in R and have centers on the x-axis. Let C be …

Let $O$ be the centre of the circle $x^2+y^2=r^2$, where $r>\frac{\sqrt{5}}{2}$. Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2x+4y=5$. If the centre of the circumcircle of the triangle $OPQ$ lies on the …

A line $\mathrm{y}=\mathrm{mx}+1$ intersects the circle $(\mathrm{x}-3{)}^2+(\mathrm{y}+2{)}^2=25$ at the points P and Q. If the midpoint of the line segment PQ has x-coordinate $-\frac{3}{5}$, then which one of the following options is correct?

Let the point B be the reflection of the point $\mathrm{A}(2,3)$ with respect to the line $8\mathrm{x}-6\mathrm{y}-23=0$. Let ${\Gamma }_{\mathrm{A}}$ and ${\Gamma }_{\mathrm{B}}$ be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles …

Answer the following by appropriately matching the lists based on the information given in the paragraph
Let the circles ${\mathrm{C}}_1:{\mathrm{x}}^2+{\mathrm{y}}^2=9$ and ${\mathrm{C}}_2:(\mathrm{x}-3{)}^2+(\mathrm{y}-4{)}^2=16$, intersect at the points X and Y. Suppose t …

Answer the following by appropriately matching the lists based on the information given in the paragraph
Let the circles ${\mathrm{C}}_1:{\mathrm{x}}^2+{\mathrm{y}}^2=9$ and ${\mathrm{C}}_2:(\mathrm{x}-3{)}^2+(\mathrm{y}-4{)}^2=16$, intersect at the points X and Y. Suppose t …

Let $S$ be the circle in the $xy$-plane defined by the equation $x^2+y^2=4$.

Let $E_1{E}_2$ and $F_1{F}_2$ be the chords of $S$ passing through the point $P_0(1,1)$ and parallel to the $x$-axis and the $y$-axis, respectively. Let $G_1{G}_2$ be …

Let $S$ be the circle in the $xy$-plane defined by the equation $x^2+y^2=4$.

Let $P$ be a point on the circle $S$ with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at the points $M$ and $N$. Then, the mid-poin …

Let $T$ be the line passing through the points $P(-2,7)$ and $Q(2,-5)$. Let $F_1$ be the set of all pairs of circles $(S_1,S_2)$ such that $T$ is tangent to $S_1$ at $P$ and tangent to $S_2$ at $Q$, and also such that $S_1$ and $S_2$ touch e …

For how many values of $p$, the circle $x^2+y^2+2x+4y-p=0$ and the coordinate axes have exactly three common points?

| (II) $x^2+a^2{y}^2=a^2$ | (ii) $y=mx+a\sqrt{m^2+1}$ | (Q) …

The circle $C_1:x^2+y^2=3$, with centre at $O$, intersects the parabola $x^2=2y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_1$ at $P$ touches other two circles $C_2$ and $C_3$ at $R_2$ and $R_3$, respectively. Suppose $C_2$ and …

Let $RS$ be the diameter of the circle $x^2+y^2=1$, where $S$ is the point $(1,0)$. Let $P$ be a variable point (other than $R$ and $S$) on the circle and tangents to the circle at $S$ and $P$ meet at the point $Q$. The normal to the circle at $P$ intersects a line drawn thro …

A circle $S$ passes through the point $(0,1)$ and is orthogonal to the circles $(x-1{)}^2+y^2=16$ and $x^2+y^2=1$. Then

The common tangents to the circle $x^2+y^2=2$ and the parabola $y^2=8x$ touch the circle at the points $P,Q$ and the parabola at the points $R,S$. Then the area of the quadrilateral $PQRS$ is

Circle(s) touching $x$-axis at a distance 3 from the origin and having an intercept of length $2\sqrt{7}$ on $y$-axis is (are)

A tangent $PT$ is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3},1)$. A straight line $L$, perpendicular to $PT$ is a tangent to the circle $(x-3{)}^2+y^2=1$.

A possible equation of $L$ is

A tangent $PT$ is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3},1)$. A straight line $L$, perpendicular to $PT$ is a tangent to the circle $(x-3{)}^2+y^2=1$.

A common tangent of the two circles is

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x-5y=20$ to the circle $x^2+y^2=9$ is

The circle passing through the point $(-1,0)$ and touching the $y$-axis at $(0,2)$ also passes through the point

The straight line $2x-3y=1$ divides the circular region $x^2+y^2\le 6$ into two parts. If

then the num …

Two parallel chords of a circle of radius 2 are at a distance $\sqrt{3}+1$ apart. If the chords subtend at the center, angles of $\frac{\pi }{\mathrm{k}}$ and $\frac{2\pi }{\mathrm{k}}$, where $\mathrm{k}>0$, then the value of $[\mathrm{k}]$ is

[Note : [k] denotes the l …

The circle $x^2+y^2-8x=0$ and hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ intersect at the points A and B.

Equation of the circle with AB as its diameter is

The centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance of 6 units from each other. Let $P$ be the mid point of the line segment joining the centres of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ extern …

Tangents drawn from the point $P(1,8)$ to the circle

touch the circle at the points $A$ and $B$. The equation of the circumcircle of the triangle $PAB$ is

A straight line through the vertex $P$ of a triangle $PQR$ intersects the side $QR$ at the point $S$ and the circumcircle of the triangle $PQR$ at the point $T$. If $S$ is not the centre of the circumcircle, then

A circle $C$ of radius 1 is inscribed in an equilateral triangle $PQR$. The points of contact of $C$ with the sides $PQ,QR,RP$ are $D,E,F$, respectively. The line $PQ$ is given by the equation $\sqrt{3}x+y-6=0$ and the point $D$ is …

Consider

where $p$ is a real number, and $C:x^2+y^2+6x-10y+30=0$.

STATEMENT-1 : If line $L_1$ is a chord of circle $C$, then line $L_2$ is not always a diameter of circle …

Tangents are drawn from the point $(17,7)$ to the circle $x^2+y^2=169$.

STATEMENT-1 : The tangents are mutually perpendicular.

because

STATEMENT-2 : The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is …

Consider the circle $x^2+y^2=9$ and the parabola $y^2=8x$. They intersect at $P$ and $Q$ in the first and the fourth quadrants, respectively. Tangents to the circle at $P$ and $Q$ intersect the $x$-axis at $R$ and tangents to the parabola at $P$ and $Q$ …

Consider the circle $x^2+y^2=9$ and the parabola $y^2=8x$. They intersect at $P$ and $Q$ in the first and the fourth quadrants, respectively. Tangents to the circle at $P$ and $Q$ intersect the $x$-axis at $R$ and tangents to the parabola at $P$ and $Q$ …

Let $ABCD$ be a quadrilateral with area 18, with side $AB$ parallel to the side $CD$ and $AB=2CD$. Let $AD$ be perpendicular to $AB$ and $CD$. If a circle is drawn inside the quadrilateral $ABCD$ touching all the sides, then its radius is