Say I have a parabola $y=ax-bx^2$ where $a,b>0$ and $y=cx-dx^2$ where $c,d>0$.
I would like to find some sort of relationship relating $c$ and $d$ with $a$ and $b$ such that the two parabolas intersect at right angles.
Now I can see that the usual route would work, with a bit of brute force. What I mean by the 'usual' route is by solving them simultaneously, acquiring the coordinates of their intersection point, substituting this into their gradient functions, letting the product be $-1$ and working from there.
However, as you can see, this is rather laborious. Is there a better way of deriving such a result?
I am thinking something to do with constructing a quadratic whose roots are the gradients of the tangents, then letting the product of roots be $-1$. However, I cannot think of how I could construct such a quadratic.