If a,b,c,d are in proportion then prove that
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If a, b, c, d are in proportion, then prove that √(a^2+5c^2b^2+5d^2) = ab.
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If a,b,c,d are in proportion, then prove that
b
2
+5d
2
a
2
+5c
2
=
b
a
Hard
Solution
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Given a,b,c and d are in proportion
b
a
=
d
c
−−−−−−−(1)
ad=bc
a=
d
bc
b
2
+5d
2
a
2
+5c
2
=
b
a
⇒
b
2
+5d
2
(
d
bc
)
2
+5c
2
⇒
b
2
+5d
2
d
2
b
2
c
2
+5c
2
⇒
b
2
d
2
+5d
4
b
2
c
2
+5c
2
d
2
⇒
d
2
(b
2
+5d
2
)
c
2
(b
2
+5d
2
)
⇒
d
2
c
2
=
d
c
⇒
b
2
+5d
2
a
2
+5c
2
=
d
c
From (1)
⇒
d
c
=
b
a
∴
b
2
+5d
2
a
2
+5x
2
=
b
a
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