First slide

Theory of equations

Question

If the roofs of the equation $\frac{1}{x + p} + \frac{1}{x + q} = \frac{1}{r}$ are equal in magnitude and opposite' in sign, then the product of root, is

Moderate

A

$- \frac{1}{2} (p^{2} - q^{2})$
B

$p^{2} + q^{2}$
C

$\frac{1}{2} (p^{2} - q^{2})$
D

$- \frac{1}{2} (p^{2} + q^{2})$

Solution

The equation $\frac{1}{x + p} + \frac{1}{x + q} = \frac{1}{r}$
or $x^{2} + (p + q - 2 r) x - r (p + q) + p q = 0$
Let $α, β$ be the roots of this equation. Then,
$α + β = - (p + q - 2 r)$ and $α β = - r (p + q) + p q$
It is given that $α + β = 0 .$ Therefore, $, p + q = 2 r$
$∴ α β = - r (p + q) + p q = - \frac{1}{2} (p + q)^{2} + p q = - \frac{1}{2} (p^{2} + q^{2})$

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