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Standard XII

Mathematics

Summation by Sigma Method

If r be the...

Question

If $r$ be the ratio of the roots of the equation $a x^{2} + b x + c = 0$ , then $\frac{{(r + 1)}^{2}}{r} =$

\frac{a^{2}}{b c}

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\frac{b^{2}}{c a}

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\frac{c^{2}}{a b}

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None of these

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Solution

The correct option is B $\frac{b^{2}}{c a}$
Given equation is $a x^{2} + b x + c = 0$ ...(1)
Let the root of the equation (1) be $α$ and $r α$ , then
$α + r α = - \frac{b}{a}$ ...(2)
and $r α^{2} = \frac{c}{a}$ ...(3)
From (2) $α = - \frac{b}{a (r + 1)}$
Putting the value of $α$ in (3), we get
$\frac{r b^{2}}{a^{2} {(r + 1)}^{2}} = \frac{c}{a} \Rightarrow \frac{b^{2}}{a c} = \frac{{(r + 1)}^{2}}{r}$

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If r be the ratio of the roots of the equation ax2+bx+c=0, then (r+1)2r=

The correct option is B b2caGiven equation is ax2+bx+c=0 ...(1)Let the root of the equation (1) be α and rα, then α+rα=−ba ...(2)and rα2=ca ...(3)From (2) α=−ba(r+1)Putting the value of α in (3), we getrb2a2(r+1)2=ca⇒b2ac=(r+1)2r

If $r$ be the ratio of the roots of the equation $a x^{2} + b x + c = 0$ , then $\frac{{(r + 1)}^{2}}{r} =$