At any point P on the parabola y^2 - 2y - 4x + 5 = 0, a tangent is drawn which meets the the directrix at Q. Then, the locus of R which divides QP externally in the ratio 1/2 : 1 is
1) (x+1) (y-1)^2 = 4
2) (x+1) (y-1)^2 + 4 = 0
3 ) (x-1) (y+1)^2 = 4
4) (x-1) ( y+1)^2 + 4 = 0
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ans = (b)
I will write the important steps.
Parabola:
(y-1)^2 = 4 a (x-1), a = 1 --------- (1)
Vertex=(1,1), Focus: (1+a,1) = (2,1).
Directrix : x= 1-a = 0 So it is the Y axis.
Slope of tangent : dy/dx = 2/(y-1) = 1/sqrt(x-1)
Let P=(x1,y1)
Tangent: y-y1 = 2 (x-x1)/ (y1-1) ------- (2)
Find Q by putting x =0 in (2)...
So Q = [0, (y1^2 - 5) / 2(y1-1) ]
Let R=(x,y)...
Q is the midpoint of PR..as QR=PR/2
So we get x = - x1,
y = (y1-5) / (y1-1) = 1 - 4/(y1-1)
Eliminate x1, y1 using (y1 - 1)^2 = 4 (x1-1)
Finally: (y-1)^2 (x+1)+4 = 0
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