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Indices (Exponents) — MATHEMATICS STD 9 — Question

ICSE BoardEnglish MediumSTD 9MATHEMATICSIndices (Exponents)3 Marks
Question
If $\mathrm{a}^{\mathrm{x}}=\mathrm{b}^{\mathrm{y}}=\mathrm{c}^{\mathrm{z}}$ and $\mathrm{b}^2=\mathrm{ac},$prove that $\mathrm{y}=\frac{2 a z}{x+z}$
✓

Answer

Let $a x=b y=c z=k$
$\therefore a=k^{\frac{1}{x}} ; b=k^{\frac{1}{y}} ; c=k^{\frac{1}{z}}$
Also, We have $b2 = ac$
$\therefore\left(k^{\frac{1}{y}}\right)^2=\left(k^{\frac{1}{x}}\right) \times\left(k^{\frac{1}{2}}\right)$
$ \Rightarrow k^{\frac{2}{y}}=k^{\frac{1}{x}+\frac{1}{z}}$
$ \Rightarrow k^{\frac{2}{y}}=\frac{k^{z+x}}{x z}$
Comparing the powers we have
$\frac{2}{y}=\frac{z+x}{x z}$
$ \Rightarrow y=\frac{2 x z}{z+x}$

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