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10-2. Parabola, Ellipse, Hyperbola
hard
The locus of the mid points of the chords of the hyperbola $\mathrm{x}^{2}-\mathrm{y}^{2}=4$, which touch the parabola $\mathrm{y}^{2}=8 \mathrm{x}$, is :
A
$\mathrm{y}^{3}(\mathrm{x}-2)=\mathrm{x}^{2}$
B
$x^{3}(x-2)=y^{2}$
C
$\mathrm{y}^{2}(\mathrm{x}-2)=\mathrm{x}^{3}$
D
$x^{2}(x-2)=y^{3}$
(JEE MAIN-2021)
Solution
$\mathrm{T}=\mathrm{S}_{1}$
$\mathrm{xh}-\mathrm{yk}=\mathrm{h}^{2}-\mathrm{k}^{2}$
$\mathrm{y}=\frac{\mathrm{xh}}{\mathrm{k}}-\frac{\left(\mathrm{h}^{2}-\mathrm{k}^{2}\right)}{\mathrm{k}}$
this touches $y^{2}=8 x$ then $c=\frac{a}{m}$
$\left(\frac{\mathrm{k}^{2}-\mathrm{h}^{2}}{\mathrm{k}}\right)=\frac{2 \mathrm{k}}{\mathrm{h}}$
$2 \mathrm{y}^{2}=\mathrm{x}\left(\mathrm{y}^{2}-\mathrm{x}^{2}\right)$
$\mathrm{y}^{2}(\mathrm{x}-2)=\mathrm{x}^{3}$
Standard 11
Mathematics
