The length of the diameter of the circle which touches the $x-$axis at the point $(1,0)$ and passes through the point $(2,3)$ is:

Suppose $S_1$ and $S_2$ are two unequal circles,$AB$ and $CD$ are the direct common tangents to these circles. $A$ transverse common tangent $PQ$ cuts $AB$ in $R$ and $CD$ in $S$. If $AB=10$,then $RS$ is

The sum of the minimum and maximum distances of the point $(4,-3)$ to the circle $x^2+y^2+4x-10y-7=0$ is

The power of a point $(2, -1)$ with respect to a circle $C$ of radius $4$ is $9$. The centre of the circle $C$ lies on the line $x+y=0$ and in the $2^{\text{nd}}$ quadrant. If $(\alpha, \beta)$ is the centre of the circle $C$,then $\beta-\alpha=$

For the two circles $x^2 + y^2 = 16$ and $x^2 + y^2 - 2y = 0$,there is/are:

The centre of the circle passing through the point $(0, 1)$ and touching the curve $y = x^2$ at $(2, 4)$ is