If the lines $3x - 4y + 4 = 0$ and $6x - 8y - 7 = 0$ are tangents to the same circle,then the area of that circle (in sq. units) is

Let $S$ be the set of all points $\left(\frac{a}{b}, \frac{c}{d}\right)$ on the circle with radius $1$ centered at $(0,0)$,where $a$ and $b$ are relatively prime integers,$c$ and $d$ are relatively prime integers (that is,$\operatorname{HCF}(a, b) = \operatorname{HCF}(c, d) = 1$),and the integers $b$ and $d$ are even. Then,the set $S$ is:

The image of the point $(3, 4)$ with respect to the radical axis of the circles $x^2 + y^2 + 8x + 2y + 10 = 0$ and $x^2 + y^2 + 7x + 3y + 10 = 0$ is

Let a circle $C$ pass through the points $(4, 2)$ and $(0, 2)$,and its centre lie on the line $3x + 2y + 2 = 0$. Then the length of the chord of the circle $C$ whose midpoint is $(1, 2)$ is:

The equation of the circle which touches the circle $S \equiv x^2+y^2-10x-4y+19=0$ at the point $(2,3)$ internally and has a radius equal to half of the radius of the circle $S=0$ is:

$AB$ is a line segment of length $24 \ cm$ and $C$ is its midpoint. On $AB$,two semicircles with diameters $AC$ and $CB$ are described on the same side. $A$ large semicircle with diameter $AB$ is also described on the same side. Find the radius of the circle that touches all three semicircles. (in $cm$)