10. A square PQRS length of its side equal to 3 + \sqrt{}5. Let M be the mid-point of the side RS. Also, let C1 be the in-circle of triangle PMS and C2 be the circle that touches thesides P Q, QR and PM. Find the radius of the circle C2.

Q. Let $ABCD$ be a square of side of unit length. Let a circle $C_1$ 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 $C_2$ meet the side $AB$ at $E$. If the length of $EB$ is $\alpha +\sqrt{3}\beta$ where $\alpha ,\beta$ are integers, then $\alpha +\beta$ is equal to

Q. Let ABCD be a square of sides of sides of length $2$ units. $C_1$ is the circle touching all the sides of the square and $C_2$ is the circle circumscribing the square. $l$ is a line through A.
(i) If P is a point on $C_1$ and Q is a point on $C_2$ then $\frac{P{A}^2+P{B}^2+P{C}^2+P{D}^2}{Q{A}^2+Q{B}^2+Q{C}^2+Q{D}^2}$ equals

Q. 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 $C_2$ extemally. 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 ofthe circle $C$ is

Q. 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 $C_1$ 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:

Q. In the adjoining figure, $ABC$ is a triangle in which $\angle B={90}^{\circ }$ and its incircle $C_1$ has radius $3$. A circle $C_2$ of radius $1$ touches sides $AC,BC$ and the circle $C_1$. Then length $AB$ is equal to