Tangency condition of a pencil to a parabola

Text

Find for which values of $m$ the lines of the pencil $y=mx+1$ are tangent to the parabola $y=x^{2}-3x+5$.

Solution

Set line and parabola equal and require the resolving equation to have a single solution: $x^{2}-3x+5=mx+1\;\Rightarrow\; x^{2}-(3+m)x+4=0.$ Tangency corresponds to a zero discriminant: $\Delta=(3+m)^{2}-16=0\;\Rightarrow\;(3+m)^{2}=16\;\Rightarrow\; 3+m=\pm 4.$ Hence $m=1$ or $m=-7$. $\boxed{\,m=1\quad\lor\quad m=-7\,}$