Yr 11 Cambridge Geometrical Theorems about the Parabola question (1 Viewer)

[cookieeater1234]

P is a variable point on the parabola x^2=4y. The normal at P meets the parabola again at Q. The tangents at P and Q meet at T. S is the focus and QS=2PS. Prove that angle PSQ is a right angle.

I thought about proving it using the reflection property and then using congruent triangles but I always come up with something where I have to assume PSQ=90 which defeats the whole point of proving it.

 

[deterministic]

(1) Let P be (2p, p^2) and Q be (2q, q^2). Find q in terms of p by using the intersection between the normal at P and parabola.

(2) remember PS=distance to focus = distance to directrix (defn of a parabola), which can easily be found by drawing a diagram. Do the same for QS in terms of p.

(3) use the relationship QS=2(PS) to solve for p

(4) Now you can easily show that QS is perpendicular to PS by showing:
gradient(QS)*gradient(PS)=-1

There could be a better way of doing all this though...