Question

If $a^x = b^y = c^z$ and $b^2 = ac$, then the value of $y$ is equal to:

• A. $\fracxzx + z$
• B. $\frac2xzx + z$
• C. $\fracxz2(x - z)$
• D. $\fracxz2(z - x)$

Hint:

When $a^x = b^y = c^z$, substitute each variable in terms of a common base and exponent to simplify the comparison.

Answer 1

The Correct Option is B

We are given:
\[ a^x = b^y = c^z = k \text(let the common value be k\text) \] From $a^x = k$ we get $a = k^1/x$
From $b^y = k$ we get $b = k^1/y$
From $c^z = k$ we get $c = k^1/z$
Also, $b^2 = ac$ is given. Substituting these values:
\[ \left(k^1/y\right)^2 = k^1/x k^1/z \] Simplifying the left-hand side:
\[ k^2/y = k^1/x + 1/z \] Since bases are the same, equate exponents:
\[ \frac2y = \frac1x + \frac1z \] Taking LCM:
\[ \frac2y = \fracz + xxz \] Inverting:
\[ y = \frac2xzx + z \] Thus, $\mathbfy = \frac2xzx + z$.