If the number of common tangents to the pair of circles $x^2+y^2-2x+4y-4=0$ and $x^2+y^2+4x-4y+\alpha=0$ is $4$,then the least integral value of $\alpha$ is

Let $B$ be the centre of the circle $x^{2}+y^{2}-2x+4y+1=0$. Let the tangents at two points $P$ and $Q$ on the circle intersect at the point $A(3,1)$. Then $8 \left(\frac{\text{Area } \triangle APQ}{\text{Area } \triangle BPQ}\right)$ is equal to:

The points of intersection of the line $ax + by = 0$ $(a \neq b)$ and the circle $x^2 + y^2 - 2x = 0$ are $A(\alpha, 0)$ and $B(1, \beta)$. The image of the circle with $AB$ as a diameter in the line $x + y + 2 = 0$ is:

Let the maximum and minimum values of $(\sqrt{8x-x^2-12}-4)^2+(x-7)^2, x \in R$ be $M$ and $m$ respectively. Then $M^2-m^2$ is equal to ...............

$A$ triangle $PQR$ is inscribed in the circle $x^2 + y^2 = 25$. If the coordinates of $Q$ and $R$ are $(3, 4)$ and $(-4, 3)$ respectively,then $\angle QPR = \dots$

Let $A, B, C$ be three points on a circle of radius $1$ such that $\angle ACB = \frac{\pi}{4}$. Then,the length of the side $AB$ is