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Byju's Answer
Standard XII
Mathematics
Chord with a Given Mid Point : Ellipse
Find the area...
Question
Find the area of the region enclosed by parabola
x
2
=
y
, the line
y
=
x
+
2
and the
x
−
a
x
i
s
.
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Solution
Let's find the point of intersection of curve
y
=
x
2
&
y
=
x
+
2
i.e., finding
A
&
B
points
x
2
=
y
⇒
x
2
=
x
+
2
⇒
x
2
−
x
−
2
=
0
⇒
(
x
−
2
)
(
x
+
1
)
=
0
∴
x
=
−
1
,
2
⇒
y
=
1
,
4
So
A
(
−
1
,
1
)
and
B
(
2
,
4
)
Required Area
=
Area of trapezium
A
B
C
D
−
Area of
A
O
B
C
D
Now, Area of trapezium
=
1
2
×
(sum of parallel sides)
×
height
=
1
2
×
(
1
+
4
)
×
3
=
15
2
units
Area under
A
O
B
C
D
is
∫
2
−
1
x
2
d
x
=
x
3
3
2
1
=
3
∴
Area required
=
15
3
−
3
=
15
−
6
2
=
9
2
units.
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