If a circle with radius $2.5$ units passes through the points $(2, 3)$ and $(5, 7)$,then its centre is

Find the circumcenter of the triangle formed by the points $(a \cos \alpha, a \sin \alpha)$,$(a \cos \beta, a \sin \beta)$,and $(a \cos \gamma, a \sin \gamma)$.

If the area of an equilateral triangle inscribed in the circle $x^2 + y^2 + 10x + 12y + c = 0$ is $27\sqrt{3} \text{ sq. units}$,then $c$ is equal to

The line $2x + y = 4$ intersects the $X$ and $Y$ axes at points $A$ and $B$ respectively. Let $C(p, q)$ be a point such that the lines $AC$ and $BC$ are perpendicular. Find the distance of point $C$ from the midpoint of segment $AB$.

If a line passing through the point $\left( -\sqrt{8}, \sqrt{8} \right)$ and making an angle of $135^{\circ}$ with the $x$-axis intersects the circle $x = 5 \cos \theta, y = 5 \sin \theta$ at points $A$ and $B$,then find the length of the chord $AB$.

The area of an equilateral triangle inscribed in the circle $x^2+y^2-6x+2y-28=0$ is . . . . . . sq. units.