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 .

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.