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**Zexuo** Problem 31 of chapter 12, section 1 of *Calculus with Analytic Geometry* by Purcell & Varberg, fifth edition:

**Prove that the tangents to a parabola at the extremities of any focal cord are perpendicular to each other.**

I solved this problem geometrically, but I first tried it algebraically and got an unexpected result:

I started with a concave up parabola with p as the distance from the vertex to the focus: $$x^2 = 4py$$

With `m` for the slope of the focal cord: $$y = mx + p$$

To determine the points on the parabola for a focal cord I substitute y in the parabola's formula: $$x^2 = 4p(mx + p) = 4pmx + 4p^2 \longrightarrow x^2 - 4pmx - 4p^2 = 0 \longrightarrow x = 2pm ± 2p\sqrt{m^2 + p^2}$$