Parabola 2 Question 19

19. At any point P on the parabola y22y4x+5=0 a tangent is drawn which meets the directrix at Q. Find the locus of point R, which divides QP externally in the ratio 12:1.

(2004,4M)

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Solution:

  1. Given equation can be rewritten as

(y1)2=4(x1), whose parametric coordinates are

x1=t2 and y1=2t

i.e. P(1+t2,1+2t)

Equation of tangent at P is,

t(y1)=x1+t2, which meets the directrix x=0 at Q.

y=1+t1t or Q0,1+t1t

Let R(h,k) which divides QP externally in the ratio

12:1 or Q is mid-point of RP.

0=h+t2+12 or t2=(h+1)

and 1+t1t=k+2t+12 or t=21k

From Eqs. (i) and (ii), 4(1k)2+(h+1)=0

or (k1)2(h+1)+4=0

Locus of a point is (x+1)(y1)2+4=0



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