Mathematics - Parabola Question with Solution | TestHub

Putting $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ in parabola ${\mathrm{y}}^2=4\mathrm{a}(\mathrm{x}+\mathrm{a})$ $\begin{array}{rlr}& \Rightarrow (\mathrm{mx}+\mathrm{c}{)}^2=4\mathrm{a}(\mathrm{x}+\mathrm{a})& \Rightarrow {\mathrm{m}}^2{\mathrm{x}}^2+2(\mathrm{mc}-2\mathrm{a})\mathrm{x}+\left({\mathrm{c}}^2-4{\mathrm{a}}^2\right)=0\end{array}$

If roots are equal i.e., $\mathrm{D}=0$ $\begin{array}{rlr}& \Rightarrow 4(mc-2a{)}^2-4m^2\left(c^2-4a^2\right)=0& \Rightarrow -mc+a+a{m}^2=0\Rightarrow c=am+a/m\end{array}$ Alternative Equation of any tangent to the parabola

$\mathrm{y}=\mathrm{m}(\mathrm{x}+\mathrm{a})=\mathrm{a}/\mathrm{m}$

comparing with $\mathrm{y}=\mathrm{mx}+\mathrm{c}$

$\mathrm{c}=\mathrm{am}+\mathrm{a}/\mathrm{m}.$