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JEE Main Math Class 12 Application of Integrals Questions
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© examsnet.com
Question : 27
Total: 78
For
a
>
0
,
let the curves
C
1
:
y
2
=
a
x
and
C
2
:
x
2
=
a
y
intersect at origin
O
and a point
P
. Let the line
x
=
b
(
0
<
b
<
a
)
intersect the chord
O
P
and the
x
-axis at points
Q
and
R
, respectively. If the line
x
=
b
bisects the area bounded by the curves,
C
1
and
C
2
,
and the area of
Δ
O
Q
R
=
1
2
,
then 'a' satisfies the equation:
[Jan. 8, 2020 (I)]
x
6
−
6
x
3
+
4
=
0
x
6
−
12
x
3
+
4
=
0
x
6
+
6
x
3
−
4
=
0
x
6
−
12
x
3
−
4
=
0
Validate
Solution:
Given eqns. are,
x
2
=
a
y
and
y
2
=
a
x
After solving, we get
x
=
a
,
y
=
a
Now, coordinates of
B
is
(
a
,
a
)
and
A
is (0,0)
Now, coordinates of
Q
is
(
b
,
b
)
1
2
b
2
=
1
2
⇒
b
=
1
Area bounded by curves and
x
=
1
is
1
∫
0
(
√
a
x
1
∕
2
−
x
2
a
)
d
x
=
1
2
a
∫
0
(
√
a
x
1
∕
2
−
x
2
a
)
d
x
⇒
2
3
√
a
−
1
3
a
=
a
2
6
⇒
4
a
√
a
−
2
=
a
3
⇒
a
6
+
4
a
3
+
4
=
16
a
3
⇒
a
6
−
12
a
3
+
4
=
0
© examsnet.com
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