Question

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

• A. $\frac{a}{c}+\frac{b}{a}+\frac{a}{b}=0$
• B. $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=0$
• 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)

The correct options are
A $\frac{a}{c}+\frac{b}{a}+\frac{a}{b}=0$
C $\frac{1}{b{c}^2}+\frac{1}{a^{2}c}+\frac{1}{c{b}^2}=0$

Given : $b{x}^2+acx+b^{2}c=0$ and $c{x}^2+abx+b^{2}c=0$
Let the common root be $\alpha$, so
$b{\alpha }^2+ac\alpha +b^{2}c=0c{\alpha }^2+ab\alpha +b^{2}c=0\Rightarrow (b-c){\alpha }^2+a(c-b)\alpha =0\Rightarrow (b-c)\alpha (\alpha -a)=0\Rightarrow \alpha =0,a$
As $a,b,c$ are non zero distinct real numbers, so
$\alpha =a$
So,
$b{a}^2+a^{2}c+b^{2}c=0$
Dividing by $abc$, we get
$\frac{a}{c}+\frac{b}{a}+\frac{a}{b}=0$
On dividing by $a^2{b}^2{c}^2$, we get
$\frac{1}{b{c}^2}+\frac{1}{a^{2}c}+\frac{1}{c{b}^2}=0$