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 vertex of the parabola $(y - 1)^2 = 8(x - 1)$ is at the centre of a circle and the parabola cuts that circle at the ends of its latus rectum. Then the equation of that circle is

The area of the triangle formed by joining the origin to the points of intersection of the line $x\sqrt{5} + 2y = 3\sqrt{5}$ and the circle $x^2 + y^2 = 10$ is

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

$P$ and $Q$ are the ends of a diameter of the circle $x^2+y^2=a^2$ where $a > \frac{1}{\sqrt{2}}$. $s$ and $t$ are the lengths of the perpendiculars drawn from $P$ and $Q$ onto the line $x+y=1$ respectively. When the product $st$ is maximum,the greater value among $s$ and $t$ is

The values of the constant term in the equation of a circle passing through $(1, 2)$ and $(3, 4)$ and touching the line $3x + y - 3 = 0$ are