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JEE Advanced 2018 Paper 2
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© examsnet.com
Question : 40
Total: 54
Consider two straight lines, each of which is tangent to both the circle
x
2
+
y
2
=
1
2
and the parabola
y
2
=
4
x
.Let these lines intersect at the point Q. Consider the ellipse whose center is at the origin O(0, 0) and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is
√
2
,then the which of the following statement(s) is (are) TRUE?
For the ellipse, the eccentricity is
1
√
2
and the length of the latus rectum is 1
For the ellipse, the eccentricity is
1
2
and the length of the latus rectum is
1
2
The area of the region bounded by the ellipse between the lines
x
=
1
√
2
and
x
=
1
is
1
4
√
2
(
π
−
2
)
The area of the region bounded by the ellipse between the lines
x
=
1
√
2
and
x
=
1
is
1
16
(
π
−
2
)
Validate
Solution:
Let equation of common tangent is
y
=
m
x
+
1
m
∴
|
0
+
0
+
1
m
√
1
+
m
2
|
=
1
√
2
⇒
m
4
+
m
2
−
2
=
0
⇒
m
±
1
Equation of common tangents are
y
=
x
+
1
and
y
=
−
x
−
1
Point Q is
(
−
1
,
0
)
∴ Equation of ellipse is
x
2
1
+
y
2
1
2
=
1
A)
e
=
√
1
+
1
2
=
1
2
and
L
R
=
2
b
2
a
=
1
C) Area
2.
1
∫
1
√
2
1
√
2
.
√
1
+
x
2
d
x
=
√
2
[
x
2
√
1
−
x
2
9
1
2
s
i
n
−
1
x
]
1
√
2
1
=
√
2
[
π
4
−
(
1
4
+
π
8
)
]
=
√
2
(
π
8
−
1
4
)
=
π
−
2
4
√
2
Correct answer are (A) and (D)
© examsnet.com
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