First slide

Theory of equations

Question

Let $α$ and $β$ the roots of the equation $p x^{2} + q x + r = 0, p \neq 0$ If $p, q, r$ are in A .P and $\frac{1}{α} + \frac{1}{β} = 4,$ ,then the value of $| α - β |$ is

Moderate

A

$\frac{\sqrt{34}}{9}$
B

$\frac{2 \sqrt{13}}{9}$
C

$\frac{\sqrt{61}}{9}$
D

$\frac{2 \sqrt{17}}{9}$

Solution

Since $α$ and $β$ pare roots of the equation
$p x^{2} + q x + r = 0 .$ Therefore $α + β = - \frac{9}{p}$ and $α β = \frac{r}{p}$
Now, $\frac{1}{α} + \frac{1}{β} = 4 \Rightarrow \frac{α + β}{α β} = 4 \Rightarrow - \frac{q}{r} = 4 \Rightarrow q = - 4$
It is given that $p, q, r$ are in A.P . Therefore,
$2 q = p + r \Rightarrow - 8 r = p + r \Rightarrow p = - 9 r$
$∴ α + β = - \frac{q}{n} = - \frac{4}{9}$ and $α β = \frac{r}{p} = - \frac{1}{y}$
Now, $(α - β)^{2} = (α + β)^{2} - 4 α β$
$\Rightarrow (α - β)^{2} = \frac{16}{81} + \frac{4}{9} = \frac{52}{81} \Rightarrow | α - β | = \frac{2 \sqrt{13}}{9}$

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