Question

If from every point on the tangent to the parabola $y^2=12x$ 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 $3x^2-a{y}^2-bx-4y=0$. Find the value of $(a+b)$.

Answer 1

Answer: 8

Upper end of latus-rectum of the parabola $y^2=12x$ is $(3,6)$
Equation of tangent at $(3,6)$ to the parabola $y^2=12x$ 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)
$\frac{x_1}{h}=\frac{y_1}{k}=\frac{1}{\frac{h^2}{4}-\frac{k^2}{1}}$

$\Rightarrow x_1=\frac{h}{\frac{h^2}{4}-\frac{k^2}{1}},y_1=\frac{k}{\frac{h^2}{4}-\frac{k^2}{1}}$
$\therefore x_1-y_1+3=0$
$h-k+3\left(\frac{h^2}{4}-\frac{k^2}{1}\right)=0$
$3x^2-12y^2+4x-4y=0$
$\therefore a+b=12-4=8$