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Q.
The equation whose roots are reciprocal of the roots of the equation $x^{2} - 5x + 6 = 0$ is
Complex Numbers and Quadratic Equations
Solution:
The roots of the equation are $2, 3$
$\therefore $ For new equation roots are $\frac{1}{2}$ and $\frac{1}{3}$
$\therefore $ Sum of the roots $=\frac{1}{2}+\frac{1}{3}=\frac{5}{6}$
Product of root $=\frac{1}{6}$
$\therefore $ Required equation is $x^{2}$ - (Sum of roots)x + Product of roots = 0
$\therefore x^{2}-\frac{5}{6} x+\frac{1}{6}=0$
or $6x^{2}-5x+1=0$ Alternative Solution : If roots are reciprocal, then new equation can be obtained from the given equation by replacing x to 1/x
Given equation is $x^{2}-5x+6=0 \, \dots(*)$
For new equation replace x to 1/x in $(*)$, we get
$\left(\frac{1}{x}\right)^{2}-5 \left(\frac{1}{x}\right)+6=0$ or, $6x^{2}-5x+1=0 $ Short Cut Method :
The quadratic equation whose roots are reciprocal of the given equation can be obtained by interchanging the coefficient of $x^{2}$ and constant term
$\therefore $ Required equation is $6x^{2}-5x+1=0$