The length of the intercept on the line $4x - 3y - 10 = 0$ by the circle $x^2 + y^2 - 2x + 4y - 20 = 0$ is

Let $l > 0$ be a real number,$C$ denote a circle with circumference $l$,and $T$ denote a triangle with perimeter $l$. Then:

Let $R$ be a rectangle,$C$ be a circle,and $T$ be a triangle in the plane. The maximum possible number of points common to the perimeters of $R, C$ and $T$ is

The coordinates of the centre of the smallest circle passing through the origin and having $y=x+1$ as a diameter are

$P(3,1)$,$Q(6,5)$ and $R(x,y)$ are three points such that the angle $\angle PRQ$ is a right angle and the area of the $\Delta RPQ = 5$. Then the number of such points $R$ is

Let $PQ$ be a diameter of the circle $x^{2}+y^{2}=9$. If $\alpha$ and $\beta$ are the lengths of the perpendiculars from $P$ and $Q$ on the straight line $x+y=2$ respectively,then the maximum value of $\alpha \beta$ is