Parametrics - Proving Questions (1 Viewer)

[j1mmy_]

Two point P (2ap, ap²) and Q (2aq, aq²) lie on the parabola x² = 4ay

a) You are given that the tangents at P and Q intersect at an angle of 45 degrees. Show that p - q = 1 + pq

b) By evaluating the expression x² - 4ay at T, or otherwise, find the locus of T when the tangents at P and Q intersect as given in a.

Already found:
Equation of tangent at parabola P: y = px - ap²
Similarly at Q, y = qx -aq²
Point of intersection of tangents P & Q - [A(p+q), apq]

 

[Carrotsticks]

a) The gradient of the tangent at P is p. The gradient of the tangent at Q is q. Using the 'angle between two lines' formula, we have tan(45) = (p-q)/(1+pq) and the result follows since tan(45) = 1.

b) Let x = A(p+q) and y = apq.

We have p^2 + q^2 - 2pq = 1 + 2pq + p^2 q^2 by squaring both sides of (a).

Completing the square of the left hand side, we have (p+q)^2 -4pq = 1 + 2pq + p^2 q^2 and re-arranging that, we have (p+q)^2 = 1 + 6pq + p^2 q^2

So (x/a)^2 = 1 + 6(y/a) + (y/a)^2

Then simplify etc etc.

 

[j1mmy_]

a) The gradient of the tangent at P is p. The gradient of the tangent at Q is q. Using the 'angle between two lines' formula, we have tan(45) = (p-q)/(1+pq) and the result follows since tan(45) = 1.

b) Let x = A(p+q) and y = apq.

We have p^2 + q^2 - 2pq = 1 + 2pq + p^2 q^2 by squaring both sides of (a).

Completing the square of the left hand side, we have (p+q)^2 -4pq = 1 + 2pq + p^2 q^2 and re-arranging that, we have (p+q)^2 = 1 + 6pq + p^2 q^2

So (x/a)^2 = 1 + 6(y/a) + (y/a)^2

Then simplify etc etc.

Okay, thank you!

What do I need to do in this question to prove this:
The tangent at the parabola x² has point P(2ap,ap²). S is the focus of the parabola, and T is the point of the intersection of the tangent and the y axis. Show that angle SPT is equal to the acute angle between the tangent and the line through P parallel to the axis of the parabola.

I've already proven that SP = ST, now I'm stuck on proving about the angle.

 

[Carrotsticks]

Prove a pair of congruent triangles first (play around with the geometry and you should see it) and use the SP = ST as one of the reasons for congruency.