$S_1$ and $S_2$ are two concentric circles of radii $1$ and $2$ respectively. Two parallel tangents to $S_1$ cut off an arc from $S_2$. The length of the arc is
$\frac{\pi }{2}$
$\frac{2\pi }{3}$
$\frac{3\pi }{4}$
$\frac{\pi }{4}$
The equations of the tangents to the circle ${x^2} + {y^2} = 13$ at the points whose abscissa is $2$, are
Two circles each of radius $5\, units$ touch each other at the point $(1,2)$. If the equation of their common tangent is $4 \mathrm{x}+3 \mathrm{y}=10$, and $\mathrm{C}_{1}(\alpha, \beta)$ and $\mathrm{C}_{2}(\gamma, \delta)$, $\mathrm{C}_{1} \neq \mathrm{C}_{2}$ are their centres, then $|(\alpha+\beta)(\gamma+\delta)|$ is equal to .... .
If the line $3x + 4y - 1 = 0$ touches the circle ${(x - 1)^2} + {(y - 2)^2} = {r^2}$, then the value of $r$ will be
The slope of the tangent at the point $(h,h)$ of the circle ${x^2} + {y^2} = {a^2}$ is
The line $y = mx + c$ will be a normal to the circle with radius $r$ and centre at $(a, b)$, if