First slide

Theory of equations

Question

If one root of the equation ${ax}^{2} + bx + c = 0$ be n times the other root, then

Moderate

A

${na}^{2} = bc {(n + 1)}^{2}$
B

${nb}^{2} = ac {(n + 1)}^{2}$
C

${nc}^{2} = ab {(n + 1)}^{2}$
D

None of these

Solution

Let the roots are $α and nα$
Sum of the roots, $α + nα = - \frac{b}{a} \Rightarrow α = - \frac{b}{a (n + 1)} . . . . . . . . . . . . . . . . . . . . . (i) and product, α . nα = \frac{c}{a} \Rightarrow α^{2} = \frac{c}{na} . . . . . . . . . . . . . . . . . . (ii) From (i) and (ii), we get \Rightarrow {[- \frac{b}{a (n + 1)}]}^{2} = \frac{c}{na} \Rightarrow \frac{b^{2}}{a^{2} {(n + 1)}^{2}} = \frac{c}{na} \Rightarrow {nb}^{2} = ac {(n + 1)}^{2}$

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