Mathematics - Parabola Question with Solution | TestHub

Let the two tangents to the parabola $y^2=4ax$ be $PT$ and $QT$ which are at right angle to one another at $\mathrm{T}(\mathrm{h},\mathrm{k})$. Then we have a find the locus of $\mathrm{T}(\mathrm{h},\mathrm{k})$. We know that $y=mx+\frac{a}{m}$, where $m$ is the slope is the equation of tangent to the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ for all $\mathrm{m}$. Since this tangent to the parabola will pass through $\mathrm{T}(\mathrm{h},\mathrm{k})$, so $\mathrm{k}=\mathrm{mh}+\frac{\mathrm{a}}{\mathrm{m}};\text{ or }{\mathrm{m}}^2\mathrm{h}-\mathrm{mk}+\mathrm{a}=0$ This is a quadratic equation in $m$, so will have two roots, say $m_1$ and $m_2$, then ${\mathrm{m}}_1+{\mathrm{m}}_2=\frac{\mathrm{k}}{\mathrm{h}}$, and ${\mathrm{m}}_1\cdot {\mathrm{m}}_2=\frac{\mathrm{a}}{\mathrm{h}}$ Given that the two tangents intersect at right angle so ${\mathrm{m}}_1\cdot {\mathrm{m}}_2=-1\text{ or }\frac{\mathrm{a}}{\mathrm{h}}=-1\text{ or }\mathrm{h}+\mathrm{a}=0$ The locus of $\mathrm{T}(\mathrm{h},\mathrm{k})$ is $\mathrm{x}+\mathrm{a}=0$, which is the equation of directrix.