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Algebraic Identities [NEW] — Maths STD 9 — Question

Gujarat BoardEnglish MediumSTD 9MathsAlgebraic Identities [NEW]4 Marks
Question
If $\text{a}^2-\frac{1}{\text{a}^2}=102,$ find the value of $\text{a}-\frac{1}{\text{a}}.$
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Answer

We have to find the value of $\text{a}-\frac{1}{\text{a}}$
Given $\text{a}^2-\frac{1}{\text{a}^2}=102$
Using identity (x - y)2 = x2 + y2 - 2xy
Here $\text{x}=\text{a},\ \text{y}=\frac{1}{\text{a}}$
$\Big(\text{a}-\frac{1}{\text{a}}\Big)^2=\text{a}^2+\Big(\frac{1}{\text{a}}\Big)^2-2\times\text{a}\times\frac{1}{\text{a}}$
$\Big(\text{a}-\frac{1}{\text{a}}\Big)^2=\text{a}^2+\frac{1}{\text{a}^2}-2\times\not\text{a}\times\frac{1}{\not\text{a}}$
By substituting $\text{a}^2-\frac{1}{\text{a}^2}=102$ we get 
$\Big(\text{a}-\frac{1}{\text{a}}\Big)^2=102-2$
$\Big(\text{a}-\frac{1}{\text{a}}\Big)^2=100$
$\Big(\text{a}-\frac{1}{\text{a}}\Big)\Big(\text{a}-\frac{1}{\text{a}}\Big)=10\times10$
$\Big(\text{a}-\frac{1}{\text{a}}\Big)=10$
Hence the value of $\Big(\text{a}-\frac{1}{\text{a}}\Big)$ is 10.

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