Question

Let tangents are drawn from $P(3,4)$ to the circle $x^2+y^2=a^2$ touches the circle at $A$ and $B$. If area of $△PAB=\frac{192}{25}$ sq. units, then $a=$

Answer 1

We know that
$△PAB=\frac{r{L}^3}{r^2+L^2}$
$=\frac{a\times {S_1}^{3/2}}{a^2+S_1}$ $(\because S_1=3^2+4^2-a^2=25-a^2)$
$\Rightarrow \frac{192}{25}=\frac{a(25-a^2{)}^{3/2}}{a^2+25-a^2}$
$\Rightarrow a(25-a^2{)}^{3/2}=3\cdot 2^6\Rightarrow a(25-a^2{)}^{3/2}=3\cdot (16{)}^{3/2}=3(25-9{)}^{3/2}$
$\Rightarrow a=3$