The normal at a point P on the ellipse x 2 + 4y 2 = 16 meets the x-axis at Q. If M is the mid point of the line segment PQ, then the locus of M intersects the latus rectum of the given ellipse at the points
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+ 
= 1
a = 4, b = 2
equation of normal 4x sec θ – 2y cosec θ = 12
M 
= (h, k) (say)
h = 
⇒ cos θ = 
and k = sin θ
+ k 2 = 1
locus 
+ y 2 = 1 ....(i)
for given ellipse e 2 = 1– 
= 
e = 
x = ± 4 × 
= ± 
....(ii)
solving (i) and (ii)
× 12 + y 2 = 1 ⇒ y 2 = 1 – 
= 
⇒ y = ± 
∴ required points 
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