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if cos 1 pa cos 1 qb then p2a2 kcos q2b2 sin2 wher
Question:
If
cos
−
1
(
p
a
)
+
cos
−
1
(
q
b
)
=
α
, then
p
2
a
2
+
k
cos
α
+
q
2
b
2
=
sin
2
α
where
k
is equal to:
MHT CET
Updated On:
Jun 23, 2024
(A)
−
2
p
q
a
b
(B)
2
p
q
a
b
(C)
−
p
q
a
b
(D)
p
q
a
b
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The Correct Option is
A
Solution and Explanation
Explanation:
Given,
p
2
a
2
+
k
cos
α
+
q
2
b
2
=
sin
2
α
.
.
.
.
.
.
(i)
cos
−
1
(
p
a
)
+
cos
−
1
(
q
b
)
=
α
As we know,
cos
−
1
x
+
cos
−
1
y
=
cos
−
1
(
x
y
−
1
−
x
2
⋅
1
−
y
2
)
cos
−
1
(
p
q
a
b
−
1
−
p
2
a
2
1
−
q
2
b
2
)
=
α
cos
α
=
(
p
q
a
b
−
1
−
p
2
a
2
1
−
q
2
b
2
)
p
q
a
b
−
cos
α
=
1
−
p
2
a
2
1
−
q
2
b
2
Squaring both sides, we get
(
p
q
a
b
−
cos
α
)
2
=
(
1
−
p
2
a
2
1
−
q
2
b
2
)
2
(
p
q
)
2
(
a
b
)
2
+
cos
2
α
−
2
p
q
a
b
cos
α
=
(
1
−
p
2
a
2
)
(
1
−
q
2
b
2
)
(
p
q
)
2
(
a
b
)
2
+
cos
2
α
−
2
p
q
a
b
cos
α
=
1
−
p
2
a
2
−
q
2
b
2
+
(
p
q
)
2
(
a
b
)
2
sin
2
α
=
p
2
a
2
+
q
2
b
2
−
2
p
q
a
b
cos
α
.
.
.
.
.
(ii)Comparing equation (i) and (ii), we get
k
=
−
2
p
q
a
b
Hence, the correct option is (A).
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