If the image of the point $(-4, 5)$ in the line $x + 2y = 2$ lies on the circle $(x + 4)^2 + (y - 3)^2 = r^2$,then $r$ is equal to:

The locus of the centers of the circles such that the point $(2, 3)$ is the midpoint of the chord $5x + 2y = 16$ is:

Tangents are drawn to a unit circle with centre at the origin from each point on the line $2x + y = 4$. Then the equation to the locus of the middle point of the chord of contact is

If the line $x \cos \theta + y \sin \theta = 2$ is the equation of a transverse common tangent to the circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 6 \sqrt{3} x - 6y + 20 = 0$,then the value of $\theta$ is:

Let $A=\{(x, y) \in R \times R \mid 2 x^{2}+2 y^{2}-2 x-2 y=1\}$,$B=\{(x, y) \in R \times R \mid 4 x^{2}+4 y^{2}-16 y+7=0\}$ and $C=\{(x, y) \in R \times R \mid x^{2}+y^{2}-4 x-2 y+5 \leq r^{2}\}$. Then the minimum value of $|r|$ such that $A \cup B \subseteq C$ is equal to:

$A$ circle passes through the point $\left( 3, \sqrt{\frac{7}{2}} \right)$ and touches the line pair $x^2 - y^2 - 2x + 1 = 0$. The coordinates of the centre of the circle are: