1.

If a^x =b^y =c^z and b^2 =ac. Prove: y =2x/x+z

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GIVEN THAT,

b {}^{2} = ac \: \: \: \: \: \: \: \: \: \: \: .....(<klux>1</klux>)

ALSO,
{a}^{x} = {b}^{y} = {<klux>C</klux>}^{z} \\ \\ let \: \\ {a}^{x} = {b}^{y} = {c}^{z} = p \\ \\ we \: have \\ {a}^{x} = p \\ \\ = > a = {p}^{ \frac{1}{x} } \: \\ \\ similarly \\ b = {p}^{ \frac{1}{y} } \: \: and \: \: c = {p}^{ \frac{1}{z} }

Putting value of a, b and c in eq(1)
We GET,

{( {p}^{ \frac{1}{y} }) }^{2} = {p}^{ \frac{1}{x} } \times {p}^{ \frac{1}{y} } \\ \\ {p}^{ \frac{2}{y} } = {p}^{ (\frac{1}{x} + \frac{1}{y} )}

On equating the power of p
We get,

\frac{2}{y} = \frac{1}{x} + \frac{1}{z} \\ \\ \frac{2}{y} = \frac{z + x}{xz} \\ \\ \frac{1}{y} = \frac{x + z}{2xz} \\ \\ = > y = \frac{2xz}{x + z} \:\:\:\:\: [Proved] \:


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