Mathematics - Parabola Question with Solution | TestHub

Solving the given equations, $$\begin{array}{rl}& (mx+c{)}^2=4ax-4a^3\\ & \Rightarrow m^2{x}^2+2mc\cdot x+c^2=4ax-4a^3\\ & \Rightarrow m^2{x}^2+(2mc-4a)x+c^2+4a^3=0\end{array}$$ Since the straight line touches the parabola at a point, so, the discriminant $=0$ $$\begin{array}{rl}& \Rightarrow (2mc-4a{)}^2-4m^2\left(c^2+4a^3\right)=0\\ & \Rightarrow 4m^2{c}^2-16amc+16a^2-4m^2{c}^2-16a^3{m}^2=0\\ & \Rightarrow -mc+a-a^2{m}^2=0\\ & \Rightarrow mc=a-a^2{m}^2\Rightarrow c=\frac{a}{m}-a^{2}m\end{array}$$