Find the equation of the circle with center $(2, 1)$ and touching the $X$-axis.

$A$ circle passes through the points $(1,2)$ and $(3,4)$. If its centre lies on the line $x-y+3=0$,then its radius is equal to

Find the equation of a circle passing through the origin and making intercepts of length $5$ on both axes.

$A$ circle has its equation in the form $x^2 + y^2 + 2x + 4y + 1 = 0$. Choose the correct coordinates of its centre and the right value of its radius from the following:

$A(2,3)$ and $B(-1,1)$ are two points. If $P(x,y)$ is a variable point such that $\angle APB = 90^{\circ}$,then the locus of $P$ is:

If the equation of the circle having its centre in the second quadrant touches the coordinate axes and also the line $\frac{x}{5}+\frac{y}{12}=1$ is $x^2+y^2+2 \lambda x-2 \lambda y+\lambda^2=0$,then $\lambda=$