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Quadratic Equations — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsQuadratic Equations1 Mark
Question
Write roots of the equation $(a − b) x^2+ (b − c) x + (c − a) = 0.$
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Answer

The given equation in $(a − b) x^2+ (b − c) x + (c − a) = 0 ....(i)$
Let $\alpha$ and $\beta$ are the roots of (i)
then, $\alpha+\beta=\frac{-\text{b}}{\text{a}}=\frac{-(\text{b}-\text{c})}{\text{a}-\text{b}}\ ...(\text{ii})$
and $\alpha\beta=\frac{\text{c}}{\text{a}}=\frac{\text{c}-\text{a}}{\text{a}-\text{b}}$
Now, $(\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta$
$\Rightarrow\Big(-\frac{\text{b}-\text{c}}{\text{a}-\text{b}}\Big)^2-4\Big(\frac{\text{c}-\text{a}}{\text{a}-\text{b}}\Big)$
$\Rightarrow\frac{(\text{b}-\text{c})^2-4(\text{c}-\text{a})(\text{a}-\text{b})}{(\text{a}-\text{b})^2}$
$\therefore\alpha-\beta=\frac{\sqrt{(\text{b}-\text{c})^2-4(\text{c}-\text{a})(\text{a}-\text{b}})}{(\text{a}-\text{b})}\ ...(\text{ii})$
Solving (i) and (ii)
$2\alpha=\frac{-(\text{b}-\text{c})}{\text{a}-\text{b}}+\frac{\sqrt{(\text{b}-\text{c})^2-4(\text{c}-\text{a})(\text{a}-\text{b}})}{\text{a}-\text{b}}$
$=\frac{-(\text{b}-\text{c})}{\text{a}-\text{b}}+\frac{\sqrt{(2\text{a}-\text{b}-\text{c})^2}}{\text{a}-\text{b}}$
$=\frac{2(\text{a}-\text{b})}{\text{a}-\text{b}}=2$
$\therefore\alpha=1$
From (ii) $\beta=\frac{\text{c}-\text{a}}{\text{a}-\text{b}}$

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