Question

If $2x+y=5$ is tangent at $(2,1)$ on hyperbola intersects asymptotes at $A$ and $B$ such that $AB=4\sqrt{5}$. If the center of the hyperbola is $(-1,-3),$ then the least possible value of sum of the semi-transverse axis and the semi-conjugate axis is $p\sqrt{q}$, where $HCF(p,q)=1$ and $q$ is a prime number, then the value of $p+q$ is

Answer 1

Any tangent to the hyperbola forms a triangle with asymptotes which has constant area $ab$.
Where $a\&b$ are the semi-transverse axis and semi-conjugate axis,
Area of $△PAB=\frac{1}{2}\times h\times AB$($\because h=$length of altitude from $P$)
$h=\frac{|-2-3-5|}{\sqrt{5}}=\frac{10}{\sqrt{5}}=2\sqrt{5}$
$Ar\left(△PAB\right)=\frac{1}{2}\times 2\sqrt{5}\times 4\sqrt{5}=20=ab$
Using $A.M\ge G.M$
$\frac{a+b}{2}\ge \sqrt{ab}a+b\ge 2\sqrt{20}=4\sqrt{5}p+q=9$