If ${c^2} > {a^2}(1 + {m^2}),$ then the line $y = mx + c$ will intersect the circle ${x^2} + {y^2} = {a^2}$

The equation of circle with centre $(1, 2)$ and tangent $x + y - 5 = 0$ is

Tangents are drawn from the point $(-1,-4)$ to the circle $x^2 + y^2 - 2x + 4y + 1 = 0$. Length of corresponding chord of contact will be-

If the tangent to the circle ${x^2} + {y^2} = {r^2}$ at the point $(a, b)$ meets the coordinate axes at the point $A$ and $B$, and $O$ is the origin, then the area of the triangle $OAB$ is

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 .... .

Let the tangents at two points $A$ and $B$ on the circle $x ^{2}+ y ^{2}-4 x +3=0$ meet at origin $O (0,0)$. Then the area of the triangle of $OAB$ is.