Question

If $x=9$ is the chord of contact of the hyperbola $x^2-y^2=9$ , then the equation of the pair of tangents forming the chord of contact is $a{x}^2-b{y}^2-18x+9=0$ . Find the value of $a+b$ .

Answer 1

Answer: 17

The equation of chord of contact at point $(h,k)$ is
$xh-yk=9$
Comparing with $x=9$ , we get
$h=1,k=0$
Equation of pair of tangents at point $(1,0)$ is
$S{S}_1=T^2$
$\Rightarrow \left(x^2-y^2-9\right)\left(1^2-0^2-9\right)={\left(x-9\right)}^2$
$\Rightarrow -8\left(x^2-y^2-9\right)=x^2-18x+81$
$\Rightarrow 9x^2-8y^2-18x+9=0$
$\Rightarrow a=9,b=8$
$\Rightarrow a+b=9+8=17$