Question

If $\text{a}$ and $\text{b}$ are the roots of equation ${\text{x}}^{\text{2}}\text{+ px + 1 = 0}\text{and}\text{c,}$ d are the roots of equation ${\text{x}}^{\text{2}}\text{+ qx + 1 = 0}$, then the value of $\text{E =}\left(\text{a - c}\right)\left(\text{b - c}\right)\left(\text{a + d}\right)\left(\text{b + d}\right)$ is

• A. $p^2-q^2$
• B. $q^2-p^2$
• C. $q^2+p^2$
• D. None of these

Answer 1

Answer: (B)

$q^2-p^2$

a and b are roots of $x^2+px+1$

$\Rightarrow a+b=-p$ and $ab1$

c and d are roots of $x^2+qx+1$

$\Rightarrow c+d=-q$ and $cd=1$

then $E=(a-c)(b-c)(a+d)(b+d)$

$=(a-c)(b+d)(b-c)(a+d)$

$=(ab+ad-bc-cd)(ab+bd-ca-cd)$

$=(1+ad-bc-1)(1+bc-ca-1)$

$=(ad-bc)(bd-ca)$

$=ab{d}^2-a^{2}cd-b^{2}cd+ab{c}^2$

$=(d^2+c^2)-(a^2+b^2)$ [$\because$ ab=1 and cd=1]

$=\left[\right(c+d{)}^2-2cd]-[(a+b{)}^2-2ab]$

$=[q^2-2]-[p^2-2]$

$=q^2-p^2$.