The length of the tangent drawn from the point $(1, 5)$ to the circle $2x^2 + 2y^2 = 3$ is ...

The equation of the circle touching the circle $x^2+y^2-6x+6y+17=0$ externally and to which the lines $x^2-3xy-3x+9y=0$ are normal is

Consider a triangle $\Delta$ whose two sides lie on the $x$-axis and the line $x+y+1=0$. If the orthocenter of $\Delta$ is $(1,1)$,then the equation of the circle passing through the vertices of the triangle $\Delta$ is

Let the mirror image of a circle $c_{1}: x^{2}+y^{2}-2x-6y+\alpha=0$ in the line $y=x+1$ be $c_{2}: 5x^{2}+5y^{2}+10gx+10fy+38=0$. If $r$ is the radius of circle $c_{2}$,then $\alpha+6r^{2}$ is equal to:

The circle $4x^2+4y^2-12x-12y+9=0$

The area of the curve $x^2 + y^2 = 2ax$ is