Question

The quadratic equation whose roots are $a^2+{\beta }^2,\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$ is

• A. $a^2{c}^2{x}^2-(b^2-2ac)(a^2+c^2)x+{(b^2-2ac)}^2=0.$
• B. $a^2{c}^2{x}^2+(b^2-2ac)(a^2+c^2)x+{(b^2-2ac)}^2=0.$
• C. $a^2{c}^2{x}^2+(b^2-2ac)(a^2+c^2)x+{(b^2+2ac)}^2=0.$
• D. $a^2{c}^2{x}^2-(b^2-2ac)(a^2+c^2)x-{(b^2+2ac)}^2=0.$

Answer 1

The correct option is A $a^2{c}^2{x}^2-(b^2-2ac)(a^2+c^2)x+{(b^2-2ac)}^2=0.$

$\alpha$ and$\beta$ are roots of equation $a{x}^2+bx+c=0$
Given the root of quadratic equation are $a^2+{\beta }^2,\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$
S = sum of roots and P = product of roots
$S={\alpha }^2+{\beta }^2+\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}=({\alpha }^2+{\beta }^2)(1+\frac{1}{{\alpha }^2{\beta }^2})$
$S=\frac{b^2-2ac}{a^2}.\frac{a^2+c^2}{c^2}$
$P=({\alpha }^2+{\beta }^2).\frac{({\alpha }^2+{\beta }^2)}{{\alpha }^2{\beta }^2}=\frac{{(b^2-2ac)}^2}{a^2{c}^2}$
Equation is:
$x^2-Sx+P=0$
$a^2{c}^2{x}^2-(b^2-2ac)(a^2+c^2)x+{(b^2-2ac)}^2=0.$