ABC26GN1977 · Surds and Indices
Subject: General Aptitude · Chapter: Surds and Indices · Exam: · Marks: · Difficulty:
If $a^{x}=b^{y}=c^{z}$ and $b^{2}=a c$, then $y$ equals
(a)$\frac{x z}{x+z}$
(b)$\frac{x z}{2(x-z)}$
(c)$\frac{x z}{2(z-x)}$
(d)$\frac{2 x z}{(x+z)}$
Answer
Explanation
Let $a^{x}=b^{y}=c^{z}=k$. Then, $a=k^{\frac{1}{x}}, b=k^{\frac{1}{y}}$ and $c=k^{\frac{1}{z}}$. $$\begin{array}{ll} \therefore & b^{2}=a c \Leftrightarrow\left(k^{\frac{1}{y}}\right)^{2}=k^{\frac{1}{x}} \times k^{\frac{1}{z}} \Leftrightarrow k^{\left(\frac{2}{y}\right)}=k^{\left(\frac{1}{x}+\frac{1}{z}\right)} \\ \therefore & \frac{2}{y}=\frac{(x+z)}{x z} \Leftrightarrow \frac{y}{2}=\frac{x z}{(x+z)} \Leftrightarrow y=\frac{2 x z}{(x+z)} . \end{array}$$
Explanation as extracted from the printed page; notation may be imperfect.
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