The equation $ax^2 + 2y^2 + 2bxy + 2x - y + c = 0$ represents a circle passing through the origin,if:

If the lines $2x + 3y + 1 = 0$ and $3x - y - 4 = 0$ lie along diameters of a circle of circumference $10\pi$,then the equation of the circle is

Find the area of the circle given by the equation $(x+1)(x+2)+(y-1)(y+3)=0$.

The equation of a circle is $\operatorname{Re}(z^{2})+2(\operatorname{Im}(z))^{2}+2 \operatorname{Re}(z)=0$,where $z=x+iy$. $A$ line which passes through the center of the given circle and the vertex of the parabola $x^{2}-6x-y+13=0$ has a $y$-intercept equal to $.....$

The radius of the circle $2 x^2 + 2 y^2 - 3 x + 2 y - 1 = 0$ is $.......$ units.

If the lines $l_1x + m_1y + n_1 = 0$ and $l_2x + m_2y + n_2 = 0$ cut the axes at concyclic points,then