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)`

Answer 1

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)`