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Question
If a,b,c are in continued proportion,
show that: `("a"^2 + "b"^2)/("b"("a" + "c")` = `("b"("a" + "c"))/("b"^2 + "c"^2)`
Solution
Sine a, b, c are in continued proportion,
`"a"/"b" = "b"/"c"`
⇒ b2 = ac
Now, (a2 + b2)(b2 + c2) = (a2 + ac)(ac + c2)
= a(a + c) c(a + c)
= ac( a + c)2
= b2 ( a + c)2
⇒ (a2 + b2)(b2+ c2) = [b(a + c)] [b(a + c)]
⇒ `("a"^2+"b"^2)/("b"("a"+"c")` = `("b"("a"+"c"))/("b"^2+"c"^2)`
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