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JEE Advanced 2019 Paper 1
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© examsnet.com
Question : 41
Total: 54
Define the collections
{
E
1
,
E
2
,
E
3
,
.
.
.
.
.
.
}
of ellipses and
{
R
1
,
R
2
,
R
3
,
.
.
.
.
.
.
}
of rectangles as follows:
E
1
:
x
2
9
+
y
2
4
=
1
R
1
: rectangle of largest area, with sides parallel to the axes, inscribed in
E
1
;
E
n
ellipse
x
2
a
n
2
+
y
2
b
n
2
=
1
of largest area inscribed in
R
n
−
1
,
n
>
1
R
n
: rectangle of largest area, with sides parallel to the axes, inscribed in
E
n
,
n
>
1
Then which of the following options is/are correct?
The distance of a focus from the centre in
E
9
is
√
5
32
The length of latus rectum of
E
9
is
1
6
The eccentricities of
E
18
and
E
19
are NOT equal
N
∑
n
=
1
(
area of
R
n
)
<
24
for each positive integer N
Validate
Solution:
E
1
:
x
2
9
+
y
2
4
=
1
(
ellipse
)
,
e
=
√
5
3
R
1
:
Max. Rectangle inscribed having sides parallel to axes
For this
A
=
(
3
c
o
s
θ
,
2
s
i
n
θ
)
,
where
θ
=
π
4
i.e
A
=
(
3
√
2
,
3
√
2
)
∴ Sides
3
√
2
,
2
√
2
Area of
R
1
=
(
3
√
2
)
(
2
√
2
)
∴ If a,b are semi axes of ellipse then
a
√
2
,
b
√
2
are sides
For
E
2
:
x
2
(
3
√
2
)
2
+
y
2
(
2
√
2
)
2
=
1
i.e semi axes are
a
√
2
,
b
√
2
.
∴
E
9
:
x
2
(
3
(
√
2
)
8
)
8
+
y
2
(
2
(
√
2
)
8
)
)
⇒ distance from centre to focus of
E
9
=
a
e
=
3
(
√
2
)
8
×
√
5
3
=
√
5
16
L.L.R
=
2
b
2
a
=
2
×
(
2
(
√
2
)
8
)
)
2
3
(
√
2
)
8
=
2
×
4
256
3
16
=
1
32
×
16
3
=
1
6
Eccentricities for all ellipses are equal.
Also sum of areas
=
R
1
+
R
2
+
R
3.
.
.
.
.
∞
=
4
[
Areas sum in
Q
1
]
=
4
[
3
√
2
2
√
2
+
3
(
√
2
)
2
2
(
√
2
)
2
+
3
(
√
2
)
3
2
(
√
2
3
)
+
.
.
.
.
.
.
]
=
(
4
)
(
6
)
[
1
2
+
1
4
+
1
8
+
.
.
.
.
.
.
.
]
=
24
×
1
2
1
−
1
2
=
24
∴
N
∑
n
=
1
(
area of
R
n
)
<
24
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