Question

If equations $b{x}^2+acx+b^{2}c=0$ and $c{x}^2+abx+b^{2}c=0$ have a common root, (where $a,b,c$ are nonzero distinct real numbers), then which of the following is/are correct

• A. $\frac{a}{c}+\frac{a}{b}+\frac{b}{a}=0$
• B. $\frac{a}{b}+\frac{a}{c}+\frac{c}{a}=0$
• C. $\frac{1}{b{c}^2}+\frac{1}{a^{2}c}+\frac{1}{c{b}^2}=0$
• D. $\frac{1}{a{c}^2}+\frac{1}{b{a}^2}+\frac{1}{c{b}^2}=0$

Answer 1

Answer: (A), (C)

$\frac{1}{b{c}^2}+\frac{1}{a^{2}c}+\frac{1}{c{b}^2}=0$
$\frac{a}{c}+\frac{a}{b}+\frac{b}{a}=0$

On solving, we get common root, $\alpha =a$ condition of common root,
$a^{2}b+a^{2}c+b^{2}c=0$ ........(i)
On dividing equation (i) by $abc$, we get
$\frac{a}{c}+\frac{a}{b}+\frac{b}{a}=0$
On dividing equation (i) by $a^2{b}^2{c}^2$, we get
$\frac{1}{b{c}^2}+\frac{1}{c{b}^2}+\frac{1}{a^{2}c}=0$