Circles

Points $E$ and $F$ are given by

Equations of the sides $QR$, $RP$ are

The number of elements in the set $A\cap B\cap C$ is

Consider
$$L_1:2x+3y+p-3=0,\qquad L_2:2x+3y+p+3=0,$$
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 $C$.

and

STATEMENT-2: If line $L_1$ is a diameter of circle $C$, then line $L_2$ is not a chord of circle $C$.

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 $x^2+y^2=338$.

The ratio of the areas of the triangles $PQS$ and $PQR$ is

The radius of the circumcircle of the triangle $PRS$ is

The radius of the incircle of the triangle $PQR$ is

The differential equation $\dfrac{dy}{dx}=\dfrac{\sqrt{1-y^2}}{y}$ determines a family of circles with

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

Match the statements in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.