If the line $y = mx + c$be a tangent to the circle ${x^2} + {y^2} = {a^2}$, then the point of contact is

Let $PQ$ and $RS$ be tangents at the extremeties of the diameter $PR$ of a circle of radius $r$. If $PS$ and $RQ$ intersect at a point $X$ on the circumference of the circle, then $2r$ equals

Two tangents are drawn from a point $P$ to the circle $x^{2}+y^{2}-2 x-4 y+4=0$, such that the angle between these tangents is $\tan ^{-1}\left(\frac{12}{5}\right)$, where $\tan ^{-1}\left(\frac{12}{5}\right) \in(0, \pi)$. If the centre of the circle is denoted by $C$ and these tangents touch the circle at points $A$ and $B$, then the ratio of the areas of $\Delta PAB$ and $\Delta CAB$ is :

Let the normals at all the points on a given curve pass through a fixed point $(a, b) .$ If the curve passes through $(3,-3)$ and $(4,-2 \sqrt{2}),$ and given that $a-2 \sqrt{2} b=3,$ then $\left(a^{2}+b^{2}+a b\right)$ is equal to ..... .

If $\frac{x}{\alpha } + \frac{y}{\beta } = 1$ touches the circle ${x^2} + {y^2} = {a^2}$, then point $(1/\alpha ,\,1/\beta )$ lies on a/an

The equation of the tangent to the circle ${x^2} + {y^2} - 2x - 4y - 4 = 0$ which is perpendicular to $3x - 4y - 1 = 0$, is