If $α$ and $β$ are the roots of the equation $x^{2} + 5 x - 7 = 0$ . Then a equation with roots
$\frac{1}{α} a n d \frac{1}{β}$
is .

7 x^{2} - 5 x - 1 = 0

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7 x^{2} + 5 x + 1 = 0

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7 x^{2} - 5 x + 1 = 0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

7 x^{2} + 5 x - 1 = 0

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Solution

The correct option is A $7 x^{2} - 5 x - 1 = 0$
The equation, $x^{2} + 5 x - 7 = 0$ has roots, $α$ and $β$ $\Rightarrow α + β = - 5 a n d α \times β = - 7 . . . . . . (1]$
when the roots of a equation are given then its equation is of the form-
$x^{2}$ - (sum of its roots)x + (product of its roots)=0
$∴ e q u a t i o n w i t h r o o t s, \frac{1}{α} a n d \frac{1}{β} i s x^{2} - (\frac{1}{α} + \frac{1}{β}) x + \frac{1}{α} \times \frac{1}{β} = 0 x^{2} - (\frac{α + β}{α \times β}) x + \frac{1}{α \times β} = 0$
using [1] we get,
$7 x^{2} - 5 x - 1 = 0$

Alternatively Solution:
If the roots of the new equation are reciprocal of the original equation, then we will replace x with $\frac{1}{x}$ in the original equation.
Hence, a new equation formed will be:
$(\frac{1}{x})^{2} + 5 (\frac{1}{x}) - 7 = 0 \Rightarrow \frac{1 + 5 x - 7 x^{2}}{x^{2}} = 0 \Rightarrow 1 + 5 x - 7 x^{2} = 0 \Rightarrow 7 x^{2} - 5 x - 1 = 0$

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