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If $\alpha ,\beta$ are roots of $x^{2}+2x+5= 0$ , then the equation whose roots are $\frac{\alpha }{\beta }$ & $\frac{\beta }{\alpha }$ is

Option 1)
$5x^{2}+6x-5= 0$

Option 2)
$5x^{2}+6x+5= 0$

Option 3)
$5x^{2}-6x-5= 0$

Option 4)
$5x^{2}-6x+5= 0$

Answers (1)

best_answer

$\alpha +\beta =-2,\: \alpha \beta =5$

$S=\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{\alpha ^{2}+\beta ^{2}}{\alpha \beta }=\frac{\left ( \alpha +\beta \right )^{2}-2\alpha \beta }{\alpha \beta }=\frac{4-10}{5}$

$\Rightarrow \: S=\frac{-6}{5}$

$p= \frac{\alpha }{\beta }\cdot \frac{\beta }{\beta }=1\: \Rightarrow \, p=1$

$\therefore$ Equation is : $x^{2}-\left ( \frac{-6}{5} \right )x+1=0$

$\Rightarrow \: 5x^{2}+6x+5=0$

$\therefore$ Option (B)

To form a Quadratic Equation given the roots -

$x^{2}-Sx+P= 0$

- wherein

S = Sum of roots

P = Product of roots

Option 1)

$5x^{2}+6x-5= 0$

This is incorrect

Option 2)

$5x^{2}+6x+5= 0$

This is correct

Option 3)

$5x^{2}-6x-5= 0$

This is incorrect

Option 4)

$5x^{2}-6x+5= 0$

This is incorrect

Posted by

Himanshu

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