If $P(0,0), Q(1,0)$ and $R\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ are three given points,then the centre of the circle for which the lines $PQ, QR$ and $RP$ are the tangents is

$A$ line drawn through the point $A(5,7)$ cuts the circle $x^2+y^2-36=0$ at the points $P$ and $Q$. Then,$AP \cdot AQ=$

$A$ rectangle $ABCD$ is inscribed in a circle whose center lies on the line $3y = x + 10$. If $A$ and $B$ are the points $(-6, 7)$ and $(4, 7)$ respectively,find the area of the rectangle.

Let $OA$ be a radius of a circle with center $O$ and radius $d$. Let $B$ be a point on the circle such that $\angle AOB = \theta$ $(< \frac{\pi}{2})$. Let $D$ be a point on $OA$ such that $BD \perp OA$. Let $E$ be the mid-point of $BD$ and $F$ be a point on the arc $AB$ such that $EF \parallel OA$. Then,the ratio of the length of the arc $AF$ to the length of the arc $AB$ is

In a triangle,two vertices are $(2, 3)$ and $(4, 0)$,and its circumcentre is $(2, z)$ for some real number $z$. The circumradius is

If $(a, b)$ is the centre of the circle passing through the vertices of the triangle formed by $x+y=6, 2x+y=4$ and $x+2y=5$,then $(a, b)$ is