The diameter of the circle,whose center lies on the line $x+y=2$ in the first quadrant and which touches both the lines $x=3$ and $y=2$,is

Let a triangle $ABC$ be inscribed in the circle $x^{2} - \sqrt{2}(x+y) + y^{2} = 0$ such that $\angle BAC = \frac{\pi}{2}$. If the length of side $AB$ is $\sqrt{2}$,then the area of the $\triangle ABC$ is equal to

The area of the circle passing through the points $(5, 2), (5, -2),$ and $(1, 2)$ is (in $\pi$)

If the power of a point $(4, 2)$ with respect to the circle $x^2 + y^2 - 2\alpha x + 6y + \alpha^2 - 16 = 0$ is $9$,then the sum of the lengths of all possible intercepts made by such circles on the coordinate axes is:

If $A(-2, 1)$,$B(0, -2)$,and $C(1, 2)$ are the vertices of a triangle $ABC$,then the perpendicular distance from its circumcentre to the side $BC$ is

The equations of the circles, which touch both the axes and the line $4x+3y=12$ and have centers in the first quadrant, are