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TS EAMCET 20-July-2022 Shift 2 Solved Paper
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© examsnet.com
Question : 59
Total: 160
If
P
(
θ
)
and
Q
(
π
2
+
θ
)
are two points on the ellipse
x
2
a
2
+
y
2
b
2
=
1
and the locus of midpoint of PQ is
x
2
α
2
+
y
2
β
2
=
1
then
a
+
b
α
+
β
=
1
√
2
√
3
1
√
3
√
2
Validate
Solution:
Given, equation of ellipse is
x
2
a
2
+
y
2
b
2
=
1
Here, coordinate of
P
is
(
a
cos
θ
,
b
s
i
n
θ
)
and coordinate of
Q
is
(
a
cos
(
π
2
+
θ
)
,
b
s
i
n
(
π
2
+
θ
)
)
=
(
−
a
s
i
n
θ
,
b
cos
θ
)
Let
R
(
h
,
k
)
be the mid-point of
P
Q
.
∴
h
=
a
cos
θ
−
a
s
i
n
θ
2
,
k
=
b
s
i
n
θ
+
b
cos
θ
2
⇒
2
h
a
=
cos
θ
−
s
i
n
θ
,
2
k
b
=
s
i
n
θ
+
cos
θ
∴
4
h
2
a
2
+
4
k
2
b
2
=
2
⇒
h
2
a
2
+
k
2
b
2
=
1
2
∴
Locus of mid-point of
P
Q
is
x
2
a
2
+
y
2
b
2
=
1
2
⇒
x
2
a
2
∕
2
+
y
2
b
2
∕
2
=
1
a
+
b
α
+
β
=
a
+
b
a
√
2
+
b
√
2
=
√
2
© examsnet.com
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