Ellipse 1 Question 11

12. Let P be a point on the ellipse x2a2+y2b2=1,0<b<a. Let the line parallel to Y-axis passing through P meet the circle x2+y2=a2 at the point Q such that P and Q are on the same side of X-axis. For two positive real numbers r and s, find the locus of the point R on PQ such that PR:RQ=r:s as P varies over the ellipse.

(2001,4M)

Passage Type Questions

Passage

Let F1(x1,0) and F2(x2,0), for x1<0 and x2>0, be the foci of the ellipse x29+y28=1. Suppose a parabola having vertex at the origin and focus at F2 intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant.

(2016 Adv.)

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Answer:

Correct Answer: 12. (a)

Solution:

  1. Given, PRRQ=rs

αbsinθasinθα=rsαsbsinθs=rasinθαrαs+αr=rasinθ+bsinθsα(s+r)=sinθ(ra+bs)αα=sinθ(ra+bs)r+s

Let the coordinates of R be (h,k).

h=acosθcosθ=ha

and

k=α=(ar+bs)sinθr+s

sinθ=k(r+s)ar+bs

On squaring and adding Eqs. (i) and (ii), we get

sin2θ+cos2θ=h2a2+k2(r+s)2(ar+bs)21=h2a2+k2(r+s)2(ar+bs)2

Hence, locus of R is x2a2+y2(r+s)2(ar+bs)2=1.



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