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

Find the equation of the circle with centre $\left(\frac{1}{2}, \frac{1}{4}\right)$ and radius $\frac{1}{12}$.

Let the abscissae of the two points $P$ and $Q$ on a circle be the roots of $x^{2}-4x-6=0$ and the ordinates of $P$ and $Q$ be the roots of $y^{2}+2y-7=0$. If $PQ$ is a diameter of the circle $x^{2}+y^{2}+2ax+2by+c=0$,then the value of $(a+b-c)$ is.

If the sides of a rectangle are given by the equations $x=-2, x=6, y=-2, y=5$,then the equation of the circle,drawn on the diagonal of this rectangle as its diameter,is

Find the centre and radius of the circle $x^{2}+y^{2}-4x-8y-45=0$.

Circles are drawn through the point $(2,0)$ to cut intercepts of length $5$ units on the $X$-axis. If their centre lies in the first quadrant,then their equation is