Question

If a^x=b^y=c^x and b^2=ac prove that 1/x+1/z=2/y

Answer 1

question correction it is c^z instead of c^x.

let a^x=b^y=c^z = k (where k is a any constant).

${a}^{x} = k$

can be written as ::

$a = {k}^{ \frac{1}{x} } .......(1)$

similarly

$b = {k}^{ \frac{1}{y} } ........(2)$

and

$c = {k}^{ \frac{1}{z} } ........(3)$

and we have given in the question that

${b}^{2} = ac$

can be written as :

$\frac{b}{c} = \frac{a}{b} ..........(4)$

dividing equation 1 and 2

we have

$\frac{a}{b} = \frac{ {k}^{ \frac{1}{x} } }{ {k}^{ \frac{1}{y} } }$

$\frac{a}{b} = {k}^{ \frac{1 }{x } - \frac{1 }{y} } .........(5)$

and similarly dividing equation 2 with 3 we have :

$\frac{b}{c} = {k}^{ \frac{1}{y} - \frac{1}{z} } ..........(6)$

using equation 4 and equating equation 5 and 6 we have

${k}^{ \frac{1}{x} - \frac{1}{y} } = {k}^{ \frac{1}{y} - \frac{1}{z} }$

so we have:

$\frac{1}{x } - \frac{1}{y} = \frac{1}{y} - \frac{1}{z}$

$\frac{1}{x} + \frac{1}{z} = \frac{2}{y}$

hence proved .

HOPE THIS WILL HELP PLEASE MARK AS BRAINLIEST AND FOLLOW ME