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Test Index
Hyperbola
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Section:
Mathematics
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© examsnet.com
Question : 14
Total: 85
Consider a hyperbola
H
having centre at the origin and foci and the
x
-axis. Let
C
1
be the circle touching the hyperbola
H
and having the centre at the origin. Let
C
2
be the circle touching the hyperbola
H
at its vertex and having the centre at one of its foci. If areas (in sq. units) of
C
1
and
C
2
are
36
Ï€
and
4
Ï€
, respectively, then the length (in units) of latus rectum of
H
is
[4 Apr 2024 Shift 2]
‌
28
3
‌
14
3
‌
10
3
‌
11
3
Validate
Solution:
👈: Video Solution
Let
H
:
‌
x
2
a
2
−
‌
y
2
b
2
=
1
‌
‌
(
b
2
=
a
2
(
e
2
−
1
)
)
‌
∴
eq
n
‌
of
‌
C
1
=
x
2
+
y
2
=
a
2
‌
‌
Ar.
‌
=
36
Ï€
‌
Ï€
a
2
=
36
Ï€
‌
a
=
6
Now radius of
C
2
can be
a
(
e
−
1
)
or a(e +1
)
for
r
=
a
(
e
−
1
)
for
r
=
a
(
e
+
1
)
Ar.
=
4
Ï€
Ï€
r
2
=
4
Ï€
Ï€
a
2
(
e
−
1
)
2
=
4
Ï€
a
2
(
e
+
1
)
2
=
4
36
Ï€
(
e
−
1
)
2
=
4
Ï€
36
(
e
+
1
)
2
=
4
e
−
1
=
‌
1
3
e
+
1
=
‌
1
3
e
=
‌
4
3
−
‌
2
3
Not possible
‌
∴
b
2
=
36
(
‌
16
9
−
1
)
=
28
‌
∴
LR
=
‌
2
b
2
a
=
‌
2
×
28
6
=
‌
28
3
© examsnet.com
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