Mathematics - Parabola Question with Solution | TestHub

The equation of tangent to $y=x^2$, be $y=mx-\frac{m^2}{4}$, putting in $y=-x^2+4x-4$, we should only get one value of $x$ i.e. Discriminate must be zero. $$\begin{array}{rrr}\therefore & mx-\frac{m^2}{4}& =-x^2+4x-4\\ \Rightarrow & x^2+x(m-4)+4-\frac{m^2}{4}& =0\\ \Rightarrow & m(m-4)& =0\\ & \therefore y=0& \text{ or }m=0,4\end{array}$$ $$\therefore y=0\text{ and }y=4(x-1)\text{ are tangents. }$$