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IIT JEE Advanced 2009 Paper 2
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© examsnet.com
Question : 39
Total: 57
The normal at a point P on the ellipse
x
2
+
4
y
2
= 16 meets the x-axis at Q. If M is the midpoint of the line segment PQ, then the locus of M intersects the latus rectums of the given ellipse at the points
(
±
3
√
5
2
,
±
2
7
)
(
±
3
√
5
2
,
±
√
19
4
)
(
±
2
√
3
,
±
+
1
7
)
(
±
2
√
3
,
±
+
4
√
3
7
)
Validate
Solution:
The normal is
4x sec ϕ - 2 y cosec ϕ = 12
Now, the points Q and M are given by
Q = (3 cos ϕ , 0)
M = (α , β)
Therefore, α =
3
c
o
s
ϕ
+
4
c
o
s
ϕ
2
=
7
2
c
o
s
ϕ
⇒ cos ϕ =
2
7
α
and β = sin ϕ ;
c
o
s
2
ϕ
+
s
i
n
2
ϕ
= 1.
Therefore,
4
49
α
2
+
β
2
= 1 ⇒
4
49
x
2
+
y
2
= 1
Hence, the rectum is x = ±
2
√
3
Hence,
48
49
+
y
2
= 1 ⇒ y = ±
1
7
(
±
2
√
3
,
±
1
7
)
Hence, the locus of M intersects the latus rectum of the given ellipse at the points
© examsnet.com
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