Question

If a variable chord of $x^2–y^2=9$ touches $y^2=12x$ and locus of middle point of these chords is $x^3+{\lambda }_{1}x{y}^2+{\lambda }_2{y}^2=0$, then the value of ${\lambda }_2-{\lambda }_1$ is

Answer 1

Given $x^2–y^2=9$
Let middle point of chord $(h,k)$, then the equation of chord is
$T=S_1\Rightarrow hx-ky=h^2-k^2\Rightarrow y=\frac{h}{k}x+\left(\frac{k^2-h^2}{k}\right)\cdots \left(1\right)$
Given parabola $y^2=12x$
Equation of tangent in slope form is
$y=mx+\frac{a}{m}\Rightarrow y=mx+\frac{3}{m}\cdots \left(2\right)$
Comparing equation $\left(1\right)$ and $\left(2\right)$, we get
$m=\frac{h}{k},\frac{3}{m}=\left(\frac{k^2-h^2}{k}\right)\Rightarrow \frac{k^2-h^2}{k}=\frac{3k}{h}\Rightarrow h^3-h{k}^2+3k^2=0$
Therefore, the required locus is
$x^3-x{y}^2+3y^2=0$
Comparing with $x^3+{\lambda }_{1}x{y}^2+{\lambda }_2{y}^2=0$, we get
${\lambda }_1=-1,{\lambda }_2=3\therefore {\lambda }_2-{\lambda }_1=4$