Question

Two circle passes through $(1,2)$ and touches the co-ordinates axes . The product of radii of circles is . If $a=8-b$ and $c^2=ab-16$ (where $a,b,c,ϵR$) then the value of $(2a-b)$ is equal to

Answer 1

Equation of a circle that touches both the coordinate axis of radius ${}^{\prime }{r}^{\prime }$ is

${(x\pm r)}^2+{(y\pm r)}^2=r^2$

But, the circle passes through $(1,2)$ and hence must lie in the first quadrant.

$\therefore$ ${(x-r)}^2+{(y-r)}^2=r^2⟶\left(1\right)$

Substituting $x=1,$ $y=2$ in $\left(1\right)$ we get

${(1-r)}^2+{(2-r)}^2=r^2$

$\Rightarrow r^2-6r+5=0$

$\Rightarrow r^2-5r-r+5=0$

$\Rightarrow (r-1)(r-5)=0$

$\therefore$ Two circles are possible with radius $r=1$ & $r=5$

$\therefore$ Product of radii $=1\times 5=5$