QJEEM22Maa481: JEE Main 2022 Mathematics - Class 11 JEEM22Maa481 Solution: If for some , not all have same sign, one of the roots of the…

Let roots of

$\therefore$ $\alpha$ + $\beta$ > 0 and $\alpha$ $\beta$ > 0

Also, it has a common root with ${x^2} + 2x - 8 = 0$

$\therefore$ The common root between above two equations is 4.

$\Rightarrow 16({p^2} + {q^2}) - 8q(p + r) + {q^2} + {r^2} = 0$

$\Rightarrow (16{p^2} - 8pq + {q^2}) + (16{q^2} - 8qr + {r^2}) = 0$

$\Rightarrow {(4p - q)^2} + {(4q - r)^2} = 0$

$\Rightarrow q = 4p$ and $r = 16p$

$\therefore$ ${{{q^2} + {r^2}} \over {{p^2}}} = {{16{p^2} + 256{p^2}} \over {{p^2}}} = 272$