The number of common tangents to the circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 6x - 8y = 24$ is

The shortest distance from the line $3x + 4y = 25$ to the circle $x^2 + y^2 - 6x + 8y = 0$ is

Let a triangle $ABC$ be inscribed in the circle $x^{2} - \sqrt{2}(x+y) + y^{2} = 0$ such that $\angle BAC = \frac{\pi}{2}$. If the length of side $AB$ is $\sqrt{2}$,then the area of the $\triangle ABC$ is equal to

Let $A(1, 4)$ and $B(1, -5)$ be two points. Let $P$ be a point on the circle $(x-1)^{2} + (y-1)^{2} = 1$ such that $(PA)^{2} + (PB)^{2}$ has a maximum value. Then the points $P, A,$ and $B$ lie on:

For what values of $m$ does the circle $x^2 + y^2 = 4x + 8y + 5$ intersect the line $3x - 4y = m$ at two distinct points?

The lengths of the tangents from the point $(1,2)$ to the circles $x^2+y^2+x+y-4=0$ and $3x^2+3y^2-x-y-k=0$ are in the ratio $4:3$. Then the value of $k$ is: