Question

if a^x=b^y=c^z, b^2=ac and 1/x+1/z=a/2 then a=?

Answer 1

$a = \frac{4}{y}$

Step-by-step explanation:

$Given, \: \: \: a^x=b^y=c^z, \: \: \: and \: \: \: b^2=ac \\ Let, \: \: \: a^x=b^y=c^z = k \\ then, \: \: \: a = {k}^{ \frac{1}{x} } , \: \: b = {k}^{ \frac{1}{y} } \: \: \: and \: \: \: c = {k}^{ \frac{1}{z} } \\ Now, \: \: \: {b}^{2} = ac \\ putting \: \: values \\ ( {k}^{ \frac{1}{y} } )^{2} = {k}^{ \frac{1}{x} } \times {k}^{ \frac{1}{z} } \\ {k}^{ \frac{2}{y} } = {k}^{ \frac{1}{x} + \frac{1}{z} } \\ \frac{2}{y} = \frac{1}{x} + \frac{1}{z} \\ given, \: \: \: \frac{1}{x} + \frac{1}{z} = \frac{a}{2} \\ So, \: \: \frac{2}{y} = \frac{a}{2} \\ \: \: \: \: \: \: \: \: a = \frac{2}{y} \times 2 \\ a = \boxed{\frac{4}{y} }$