Mathematics - Hyperbola Question with Solution | TestHub

Let $\mathrm{P}(\mathrm{h},\mathrm{k})$ be midpoint of chords so their equation is $\mathrm{T}={\mathrm{S}}_1$

i.e. $xh-yk=h^2-k^2$ ..........(i) Also equation of tangent to the parabola $y^2=4ax$ is $y=mx+\frac{\mathrm{a}}{\mathrm{m}}.\dots \dots \dots \left(ii\right)$ $\therefore$ comparing (i) and (ii), we get $\mathrm{m}=\frac{\mathrm{h}}{\mathrm{k}}\text{ and }\frac{\mathrm{a}}{\mathrm{m}}=\frac{{\mathrm{k}}^2-{\mathrm{h}}^2}{\mathrm{k}}\Rightarrow \frac{\mathrm{ak}}{\mathrm{h}}=\frac{{\mathrm{k}}^2-{\mathrm{h}}^2}{\mathrm{k}}\Rightarrow {\mathrm{h}}^3={\mathrm{k}}^2(\mathrm{h}-\mathrm{a})$ Hence locus is ${\mathrm{x}}^3={\mathrm{y}}^2(\mathrm{x}-\mathrm{a})$.