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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.

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.
 

Grade:12

1 Answers

Aditya Gupta
2081 Points
4 years ago
point of intersection of tangents can be easily found out to be R(h. k) as
h= apq and k = a(p+q)
now,  tangent at P is perpendicular to the chord OQ. so prod of slopes will be – 1.
or (1/p)*(2aq/aq^2)= – 1
pq= – 2
apq= – 2a
h= – 2a
replacing h by x
reqd locus: x+2a= 0
kindly approve :))

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