Answer to Question #226664 in Analytic Geometry for Anuj

Question

The tangents at the points P(ap² , 2ap) and Q(aq² , 2aq) on the parabola y²= 4ax intersect at the point R. Given that the tangent at P is perpendicular to the chord OQ, where O is the origin, find the equation of the locus of R as p varies.

Answer 1

$P(ap^2,2ap), Q(aq^2,2aq)\\ y^2=4ax$

The tangents is

$y\cdot y_0=2a(x+x_0)$

PR:

$y\cdot 2ap=2a(x+ap^2)\\ x-py+ap^2=0$

QR:

$y\cdot 2aq=2a(x+aq^2)\\ x-qy+aq^2=0$

R=PR$\cap$ QR

$x-py+ap^2=0\\ x-qy+aq^2=0\\ y=\frac{a(p^2-q^2)}{q-p}=a(p+q)\\ x=aqp\\ R(aqp,a(p+q))$

OQ:

$\frac{x-0}{aq^2-0}=\frac{y-0}{2aq-0}\\ y=\frac{2}{q}x\\ k_{OQ}=\frac{2}{q}$

OQ$\bot$ RP

$k_{RP}=-\frac{1}{k_{OQ}}=-\frac{q}{2}$

RP:

$y=\frac{1}{p}x+ap\\ k_{RP}=\frac{1}{p}\\ \frac{1}{p}=-\frac{q}{2}\\ pq=-2$

multiply by a

$apq=-2a\\ x=-2a\\ x+2a=0$