1. ## Parametric Question

Hi

P(2ap, ap^2), Q(2aq,aq^2) and R(2ar,ar^2) are points on the parabola x^2 = 4ay.
a) Show that the equation of the normal at P is py+x = 2ap + ap^3.
b) Find the coordinates of the point of intersection of the normals at P and Q.

c) If the normals P,Q and R are constant, show that p +q +r=0.

How on earth do you do this??

2. Originally Posted by xwrathbringerx
Hi

P(2ap, ap^2), Q(2aq,aq^2) and R(2ar,ar^2) are points on the parabola x^2 = 4ay.
a) Show that the equation of the normal at P is py+x = 2ap + ap^3.
b) Find the coordinates of the point of intersection of the normals at P and Q.

c) If the normals P,Q and R are constant, show that p +q +r=0.

How on earth do you do this??

hi

(a) Use the chain rule to get the gradient of tangent , then use m1m2=-1 for the gradient of normal

$x=2ap\Rightarrow \frac{dx}{dp}=2a$

$y=ap^2\Rightarrow \frac{dy}{dp}=2ap$

so $\frac{dy}{dx}=\frac{dy}{dp}\cdot \frac{dp}{dx}=2ap\cdot \frac{1}{2a}=p$

Equation : $y-ap^2=-\frac{1}{p}(x-2ap)$
(b) Since Q is also a point on the parabola , its equation of normal would be $qy+x=aq^3+2aq$ , just swap the 'p' with 'q' , then have them equal to solve for the point of intersection .