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At any point P on the parabola y 2 – 2y – 4x + 5 =0, a tangent is drawn which meets the directrix at Q . Find the locus of R which divides QP externally in the ratio ½ : 1 .

At any point P on the parabola y2 – 2y – 4x + 5 =0, a tangent is drawn which meets the directrix at Q . Find the locus of R which divides QP externally in the ratio
½ : 1 .

Grade:12

1 Answers

Latika Leekha
askIITians Faculty 165 Points
6 years ago
The given parabola isy2- 2y – 4x + 5 = 0.
This can be rewritten as (y-1)2 = 4(x-1)
Its parametric coordinates are x-1 = t2 and y -1 = 2t and hence we have P(1 +t2, 1 + 2t)
Hence, the equation of tangent at P is
t(y-1) = x – 1 + t2, which meets the directrix x = 0 at Q.
Hence, y = 1 + t – 1/t or Q(0, 1 + t – 1/t)
Let R(h, k) be the point which divides QP externally in the ratio 1/2:1. Q is the mid-point of RP
So, we have 0 = (h + t2+ 1)/2 which gives t2 = – (h + 1) …. (1)
and 1 + t – 1/t = (k + 2t + 1)/2 which gives t = 2/(1- k) ….. (2)
Hence, from equations (1) and (2) we get, 4/(1-k)2 + (h + 1) = 0 or (k-1)2(h +1) + 4 = 0
Hence, the locus of a point is
(x + 1)(y - 1)2 = 0.

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