If the line through the point $P(5,3)$ meets the circle $x^2+y^2-2x-4y+\alpha=0$ at $A(4,2)$ and $B(x_1, y_1)$,then $PA \cdot PB$ is equal to

If $P(2, 8)$ is an interior point of a circle $x^2 + y^2 - 2x + 4y - p = 0$ which neither touches nor intersects the axes,then the set for $p$ is

Let the line $x-y+1=0$ intersect the circle $x^2+y^2+2x+2y+1=0$ at two points $A$ and $B$. If $AB$ is the diameter of the circle $x^2+y^2+2gx+2fy+c=0$,then $g+f=$

The sum of diameters of the circles that touch $(i)$ the parabola $75x^2 = 64(5y - 3)$ at the point $\left(\frac{8}{5}, \frac{6}{5}\right)$ and $(ii)$ the $y$-axis,is equal to $......$

The minimum distance between the curves $x^2 + y^2 + 4x + 16y + 66 = 0$ and $y^2 = 8x$ is:

The number of real circles cutting orthogonally the circle $x^{2}+y^{2}+2x-2y+7=0$ is