Question

The condition for the equations $a{x}^2+bx+c=0$ and $a^{\prime }{x}^2+b^{\prime }x+c^{\prime }=0$ to have reciprocal roots is $\frac{a}{c^{\prime }}=\frac{b}{b^{\prime }}=\frac{c}{a^{\prime }}$

• A. True
• B. False

Answer 1

The correct option is A True

$a{x}^2+bx+c=0$

${\alpha }_1+{\beta }_1=\frac{-b}{a}$ (equation $01$)

${\alpha }_1{\beta }_1=\frac{c}{a}$ (equation $02$)

$a^{\prime }{x}^2+b^{\prime }x+c^{\prime }=0$

${\alpha }_2+{\beta }_2=-\frac{b^{\prime }}{a^{\prime }}$ (equation $03$)

${\alpha }_2{\beta }_2=\frac{c^{\prime }}{a^{\prime }}$ (equation $04$)

Here, ${\alpha }_1=\frac{1}{{\alpha }_2}$,

${\beta }_1=\frac{1}{{\beta }_2}$

Using $01$ and $02$ we get,

$\frac{{\beta }_1}{{\alpha }_2}=\frac{c}{a}$

$\frac{{\beta }_1}{c^{\prime }}\times \frac{a^{\prime }}{{\beta }_2}=\frac{c}{a}$

$\Rightarrow \frac{{\beta }_1}{{\beta }_2}=\frac{c}{a}\times \frac{c^{\prime }}{a^{\prime }}$

$\frac{c}{a^{\prime }}=\frac{a}{c^{\prime }}$

$\therefore$ True