Tangents and normals

The set of all real values of $\lambda$ for which exactly two common tangents can be drawn to the circles $x^2+y^2-4x-4y+6=0$ and $x^2+y^2-10x-10y+\lambda=0$ is the interval :

For the two circles $x^2+y^2=16$ and $x^2+y^2-2y=0$, there is/are :

A possible equation of $L$ is

A common tangent of the two circles is

Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

Tangents drawn from the point $P(1,8)$ to the circle $$x^2+y^2-6x-4y-11=0$$ touch the circle at the points $A$ and $B$. The equation of the circumcircle of the triangle $PAB$ is

Match the conics in Column I with the statements/expressions in Column II.

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 $C_2$ externally. If a common tangent to $C_1$ and $C$ passing through $P$ is also a common tangent to $C_2$ and $C$, then the radius of the circle $C$ is

Points $E$ and $F$ are given by

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

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

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.