First slide

Theory of equations

Question

$If α, β are the roots of the equation x^{2} + p x - r = 0 and \frac{α}{3}, 3 β are the roots of the equation x^{2} + q x - r = 0, then r eqaul$

Moderate

A

$\frac{3}{8} (p - 3 q) (3 p + q)$
B

$\frac{3}{8} (p + 3 q) (3 p - q)$
C

$\frac{3}{64} (3 p - q) (p - 3 q)$
D

$\frac{3}{64} (3 q - p) (p - q)$

Solution

$Roots of the equation x^{2} + p x - r = 0 are α, β$
$∴ α + β = - p - - - - - (1)$
$and α β = - r - - - - (2)$
$Also, roots of the equation x^{2} + q x - r = 0 are \frac{α}{3} and 3 β$
$\begin{array}{l} ∴ \frac{α}{3} + 3 β = - q \\ \Rightarrow α + 9 β = - 3 q - - - - - - (3) \\ And (\frac{α}{3}) (3 β) = - r or α β = - r \end{array}$
On solving, (1) and (3), we get
$\begin{array}{l} α = \frac{3 (q - 3 p)}{8}, β = \frac{p - 3 q}{8} \\ Now α β = - r = \frac{3 (q - 3 p) (p - 3 q)}{64} \end{array}$

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