First slide

Theory of equations

Question

If A.M. of the roots of a quadratic equation is 8/5 and A.M. of their reciprocals is 8/7, then the equation is

Moderate

A

$5 x^{2} - 16 x + 7 = 0$
B

$7 x^{2} - 16 x + 5 = 0$
C

$7 x^{2} - 16 x + 8 = 0$
D

$3 x^{2} - 12 x + 7 = 0$

Solution

Let the roots are $α and β$
$\Rightarrow \frac{α + β}{2} = \frac{8}{5} \Rightarrow α + β = \frac{16}{5} . . . (i)$
and $\frac{\frac{1}{α} + \frac{1}{β}}{2} = \frac{8}{7} \Rightarrow \frac{α + β}{2 αβ} = \frac{8}{7} \Rightarrow \frac{(16 / 5)}{2 (8 / 7)} = αβ$
$\Rightarrow αβ = \frac{7}{5} . . . (ii)$
$∴ Equation is x^{2} - (\frac{16}{5}) x + \frac{7}{5} = 0 \Rightarrow 5 x^{2} - 16 + 7 = 0$

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