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Let $\alpha$ and $\beta$ be the roots of equation $px^{2}+qx+r=0,p\neq 0.\; if\; p,q,r$ are in A.P. and $\frac{1}{\alpha }+\frac{1}{\beta }=4,$ then the value of $\left | \alpha -\beta \right |$ is ?

Option 1)
$\frac{\sqrt{34}}{9}$

Option 2)
$\frac{2\sqrt{13}}{9}$

Option 3)
$\frac{\sqrt{61}}{9}$

Option 4)
$\frac{2\sqrt{17}}{9}$

Answers (2)

best_answer

As we have learned

Sum of Roots in Quadratic Equation -

$\alpha +\beta = \frac{-b}{a}$

- wherein

$\alpha \: and\beta$ are root of quadratic equation

$ax^{2}+bx+c=0$

$a,b,c\in C$

Product of Roots in Quadratic Equation -

$\alpha \beta = \frac{c}{a}$

- wherein

$\alpha \: and\ \beta$ are roots of quadratic equation:

$ax^{2}+bx+c=0$

$a,b,c\in C$

@1449

$|\alpha -\beta | = \left | \frac{\sqrt{q^2}-4pr}{p} \right |$

$\left ( \because \left | \frac{\sqrt{D}}a{} \right | \right )$

Also $\frac{\alpha +\beta }{\alpha \beta }= 4$

$\Rightarrow \frac{-q}{r}= 4$

$\Rightarrow q = -4r ....(1)$

$= \sqrt{16(\frac{r}{p})^2-(4\frac{r}{p})}$

Also p+r =2q

$\Rightarrow p+r = -8r \Rightarrow r/p = -1/9$

$\therefore \frac{\left | \alpha -\beta \right |}{16\times 1/81+4/9}= \sqrt{\frac{52}{81}}=\frac{2\sqrt{13}}{9}$

Option 1)

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

Option 2)

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

Option 3)

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

Option 4)

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

Posted by

Himanshu

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