Question

If from every point on the tangent to the parabola $y^2=12 x$ at its upper end of latus-rectum, pair of tangents are drawn to the hyperbola $\frac{x^2}{4}-\frac{y^2}{1}=1$, then locus of mid-points of their corresponding chord of contact is $3 x^2-a y^2-b x-4 y=0$. Find the value of $(a+b)$.

Answer 1

Upper end of latus-rectum of the parabola $y^2=12 x$ is $(3,6)$
Equation of tangent at $(3,6)$ to the parabola $y^2=12 x$ is
$y \cdot 6=2 \cdot 3 \cdot(x+3) $
$y=x+3$
Let $A\left(x_1, y_1\right)$ be a point on the line $x-y+3=0$ COC of hyperbola is
$\frac{x_1}{4}-\frac{y_1}{1}=1$....(i)
Equation of chord whose middle point is $(h, k)$
$\frac{ hx }{4}-\frac{ ky }{1}=\frac{ h ^2}{4}-\frac{ k ^2}{1}$....(ii)
From (i) and (ii)
$\frac{ x _1}{ h }=\frac{ y _1}{ k }=\frac{1}{\frac{ h ^2}{4}-\frac{ k ^2}{1}}$

$\Rightarrow x _1=\frac{ h }{\frac{ h ^2}{4}-\frac{ k ^2}{1}}, y _1=\frac{ k }{\frac{ h ^2}{4}-\frac{ k ^2}{1}} $
$\therefore x _1- y _1+3=0 $
$ h - k +3\left(\frac{ h ^2}{4}-\frac{ k ^2}{1}\right)=0$
$3 x ^2-12 y ^2+4 x -4 y =0$
$\therefore a + b =12-4=8$