Past exam of the mathematical methods I course of the University of Oxford / 2021 / 10 / b / Solution

The two real slope roots of $p^2+fp-1=0$ are distinct because their discriminant is $f^2+4>0$. Vieta's formulas give $p_{+}{p}_{-}=-1$, so the tangent vectors $(1,p_{+})$ and $(1,p_{-})$ have zero scalar product. Whenever the coefficients define local solution curves, the two families are therefore orthogonal trajectories.

For the stated example at $(3,4)$, the slopes are $3$ and $-1/3$. In the family from part a, the constants are $1$ and $-1/9$, givingBoth curves pass through $(3,4)$, and $3(-1/3)=-1$ verifies the right angle.