First slide

Theory of equations

Question

If the roots of the equation x² + px + q = 0 are in the same ratio as those of the equation x² + lx + m = 0, then p²m is equal to

Moderate

A

Iq
B

I²q
C

lq²
D

None of the above

Solution

Let the roots of each equation be in the ratio k:1 .
Let k $α$ and o be the roots of x² + px + g = 0 .
Then $kα + α = - p and kα \cdot α = q$
$\Rightarrow α = \frac{- p}{k + 1} and α^{2} = \frac{q}{k}$
$\begin{array}{l} \Rightarrow {(\frac{- p}{k + 1})}^{2} = \frac{q}{k} \\ \Rightarrow \frac{k}{(k + 1)^{2}} = \frac{q}{p^{2}} . . . . . . (i) \end{array}$
Again, let $kβ$ and $β$ be the roots of x² + lx+ m =0
Then, $kβ + β = - I and kβ \cdot β = m$
$\Rightarrow β = - \frac{1}{k + 1} and β^{2} = \frac{m}{k}$
$\begin{array}{ll} \Rightarrow {(\frac{- 1}{k + 1})}^{2} = \frac{m}{k} \\ \Rightarrow \frac{k}{(k + 1)^{2}} = \frac{m}{l^{2}} . . . . . . (ii) \end{array}$
Thus, from Eqs. (i) and (ii), we get
$\begin{array}{l} \frac{q}{p^{2}} = \frac{m}{l^{2}} \\ \Rightarrow p^{2} m = I^{2} q \end{array}$

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