Question

If $\alpha ,\beta$ be the roots of $a{x}^2+bx+c=0$ and $\gamma ,\delta$ those of $l{x}^2+mx+n=0$, then the equation whose roots are $\alpha \gamma +\beta \delta$ and $\alpha \delta +\beta \gamma$ is

• A. $al{x}^2+mbx+\frac{c{m}^2}{l}+\frac{n{b}^2}{a}+4cn=0$
• B. $al{x}^2-mbx+\frac{c{m}^2}{l}+\frac{n{b}^2}{a}+4cn=0$
• C. $al{x}^2-mbx+\frac{c{m}^2}{l}+\frac{n{b}^2}{a}-4cn=0$
• D. $al{x}^2+mbx+\frac{c{m}^2}{l}+\frac{n{b}^2}{a}-4cn=0$

Answer 1

The correct option is C $al{x}^2-mbx+\frac{c{m}^2}{l}+\frac{n{b}^2}{a}-4cn=0$

$a{x}^2+bx+c=0$
$\alpha +\beta =-\frac{b}{a},\alpha \beta =\frac{c}{a}$

$l{x}^2+mx+n=0$
$\gamma +\delta =-\frac{m}{l},\gamma \delta =\frac{n}{l}$

The roots are $\alpha \gamma +\beta \delta$ and $\alpha \delta +\beta \gamma$, then sum and product of the roots is,

$S=\alpha \gamma +\beta \delta +\alpha \delta +\beta \gamma =(\alpha +\beta )(\gamma +\delta )=\frac{mb}{al}$

$P=(\alpha \gamma +\beta \delta )(\alpha \delta +\beta \gamma )={\alpha }^2\gamma \delta +\alpha \beta {\delta }^2+\alpha \beta {\gamma }^2+{\beta }^2\gamma \delta =\gamma \delta ({\alpha }^2+{\beta }^2)+\alpha \beta ({\gamma }^2+{\delta }^2)=\frac{n}{l}\times \left(\frac{b^2-2ac}{a^2}\right)+\frac{c}{a}\left(\frac{m^2-2ln}{l^2}\right)=\frac{1}{al}[\frac{n{b}^2}{a}+\frac{c{m}^2}{l}-4cn]$

Therefore, the required equation is,
$x^2-Sx+P=0\therefore al{x}^2-mbx+\frac{c{m}^2}{l}+\frac{n{b}^2}{a}-4cn=0$