The number of common tangents to the circles ${x^2} + {y^2} - x = 0$ and ${x^2} + {y^2} + x = 0$ is:

The line $y=mx+c$ intersects the circle $x^2+y^2=r^2$ at two distinct points if:

Let two circles $C_1$ and $C_2$ of radii $r_1 = 2$ and $r_2 = 4$ be tangent to each other at point $P$ and tangent to a common straight line (not passing through $P$) at points $Q$ and $R$ respectively. Then the value of $PQ^2 + QR^2 + RP^2$ is:

Find the maximum distance of the point $K(10, 7)$ from the circle $x^2 + y^2 - 4x - 2y - 20 = 0$.

Observe the following statements:
$I$. The circle $x^2+y^2-6x-4y-7=0$ touches the $y$-axis.
$II$. The circle $x^2+y^2+6x+4y-7=0$ touches the $x$-axis.
Which of the following is a correct statement?

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