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JEE Main Math Class 11 Coordinate Geometry Part 4 Questions
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© examsnet.com
Question : 28
Total: 100
An ellipse
E
:
x
2
a
2
+
y
2
b
2
=
1
passes through the vertices of the hyperbola
H
:
x
2
49
−
y
2
64
=
−
1
. Let the major and minor axes of the ellipse
E
coincide with the transverse and conjugate axes of the hyperbola
H
, respectively. Let the product of the eccentricities of
E
and
H
be
1
2
. If
l
is the length of the latus rectum of the ellipse
E
, then the value of
113
l
is equal to ________.
[27-Jul-2022-Shift-1]
Your Answer:
Validate
Solution:
Vertices of hyperbola
=
(
0
,
±
8
)
As ellipse pass through it i.e.,
0
+
64
b
2
=
1
⇒
b
2
=
64
.
.
.
.
.
.
(1)
As major axis of ellipse coincide with transverse axis of hyperbola we have
b
>
a
i.e.
e
E
=
√
1
−
a
2
64
=
√
64
−
a
2
8
and
e
H
=
√
1
+
49
64
=
√
113
8
∴
e
E
⋅
e
H
=
1
2
=
√
64
−
a
2
√
113
64
⇒
(
64
−
a
2
)
(
113
)
=
32
2
⇒
a
2
=
64
−
1024
113
L.R of ellipse
=
2
a
2
b
=
2
8
(
113
×
64
−
1024
113
)
=
l
=
1552
113
∴
113
l
=
1552
© examsnet.com
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