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Section:
Mathematics
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© examsnet.com
Question : 39
Total: 102
Let a line
L
pass through the point of intersection of the lines
b
x
+
10
y
−
8
=
0
and
2
x
−
3
y
=
0
,
b
∈
R
−
{
‌
4
3
}
. If the line
L
also passes through the point
(
1
,
1
)
and touches the circle
17
(
x
2
+
y
2
)
=
16
, then the eccentricity of the ellipse
‌
x
2
5
+
‌
y
2
b
2
=
1
is :
[29-Jul-2022-Shift-1]
‌
2
√
5
√
‌
3
5
‌
1
√
5
√
‌
2
5
Validate
Solution:
L
1
:
b
x
+
10
y
−
8
=
0
,
L
2
:
2
x
−
3
y
=
0
then
L
:
(
b
x
+
10
y
−
8
)
+
λ
(
2
x
−
3
y
)
=
0
∵
It passes through
(
1
,
1
)
∴
b
+
2
−
λ
=
0
⇒
λ
=
b
+
2
and touches the circle
x
2
+
y
2
=
‌
16
17
|
‌
8
2
(
2
λ
+
b
)
2
+
(
10
−
3
λ
)
2
|
=
‌
16
17
⇒
4
λ
2
+
b
2
+
4
b
λ
+
100
+
9
λ
2
−
60
λ
=
68
⇒
13
(
b
+
2
)
2
+
b
2
+
4
b
(
b
+
2
)
−
60
(
b
+
2
)
+
32
=
0
⇒
18
b
2
=
36
∴
b
2
=
2
∴
Eccentricity of ellipse :
‌
x
2
5
+
‌
y
2
b
2
=
1
is
∴
e
=
√
1
−
‌
2
5
=
√
‌
3
5
© examsnet.com
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