Answer to Question #223997 in Analytic Geometry for Bless

Question #223997

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.


1
Expert's answer
2021-08-18T10:28:06-0400

Answer:-

P(ap2,2ap),Q(aq2,2aq)y2=4axP(ap^2,2ap), Q(aq^2,2aq)\\ y^2=4ax

The tangents is

yy0=2a(x+x0)y\cdot y_0=2a(x+x_0)

PR:

y2ap=2a(x+ap2)xpy+ap2=0y\cdot 2ap=2a(x+ap^2)\\ x-py+ap^2=0

QR:

y2aq=2a(x+aq2)xqy+aq2=0y\cdot 2aq=2a(x+aq^2)\\ x-qy+aq^2=0

R=PR\cap QR

xpy+ap2=0xqy+aq2=0y=a(p2q2)qp=a(p+q)x=aqpR(aqp,a(p+q))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:

x0aq20=y02aq0y=2qxkOQ=2q\frac{x-0}{aq^2-0}=\frac{y-0}{2aq-0}\\ y=\frac{2}{q}x\\ k_{OQ}=\frac{2}{q}

OQ\bot RP

kRP=1kOQ=q2k_{RP}=-\frac{1}{k_{OQ}}=-\frac{q}{2}

RP:

y=1px+apkRP=1p1p=q2pq=2y=\frac{1}{p}x+ap\\ k_{RP}=\frac{1}{p}\\ \frac{1}{p}=-\frac{q}{2}\\ pq=-2


multiply by a

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

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