Question

The equation whose roots are reciprocal of the roots of the equation $x^2-5x+6=0$ is

• A. $5x^2-6x+1=0$
• B. $6x^2-5x+1=0$ is
• C. $6x^2+5x-1=0$
• D. None of these

Answer 1

Answer: (B)

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 $x^2-5x+6=0\dots (\ast )$
For new equation replace x to 1/x in $(\ast )$, 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 $6x^2-5x+1=0$