# If ax = by = cz and b2

Question:

If $a^{x}=b^{y}=c^{2}$ and $b^{2}=a c$, then show that

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

Solution:

Let $a^{x}=b^{y}=c^{z}=k$

$a=k^{1 / x}, b=k^{1 / y}, c=k^{1 / z}$

Now

$b^{2}=a c$

$\left(k^{1 / y}\right)^{2}=k^{1 / x} \times k^{1 / z}$

$k^{2 / y}=k^{1 / x+1 / z}$

$2 / y=1 / x+1 / z$

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

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