$A$ square is inscribed in the circle $x^2 + y^2 - 2x + 4y + 3 = 0$,whose sides are parallel to the coordinate axes. One vertex of the square is

The number of common tangents to two circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 8x + 12 = 0$ 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 midpoint 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

If $5x - 12y + 10 = 0$ and $12y - 5x + 16 = 0$ are two tangents to a circle,then the radius of the circle is

Let $AB$ be a line segment with mid-point $C$ and $D$ be the mid-point of $AC$. Let $C_1$ be the circle with diameter $AB$ and $C_2$ be the circle with diameter $AC$. Let $E$ be a point on $C_1$ such that $EC$ is perpendicular to $AB$. Let $F$ be a point on $C_2$ such that $DF$ is perpendicular to $AB$ and $E$ and $F$ lie on opposite sides of $AB$. Then,the value of $\sin \angle FEC$ is

The radius of the circle whose arc of length $15 \ cm$ makes an angle of $3/4$ radian at the centre is ..... $cm$.