If the straight line $4x + 3y + \lambda = 0$ touches the circle $2({x^2} + {y^2}) = 5$, then $\lambda $ is

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 :

The equation of normal to the circle $2{x^2} + 2{y^2} - 2x - 5y + 3 = 0$ at $(1, 1)$ is

A line $lx + my + n = 0$ meets the circle ${x^2} + {y^2} = {a^2}$ at the points $P$ and $Q$. The tangents drawn at the points $P$ and $Q$ meet at $R$, then the coordinates of $R$ is

The two circles which passes through $(0,a)$ and $(0, - a)$ and touch the line $y = mx + c$ will intersect each other at right angle, if

If the centre of a circle is $(2, 3)$ and a tangent is $x + y = 1$, then the equation of this circle is