If equations $b{x}^2+acx+b^{2}c=0$ and $c{x}^2+abx+b^{2}c=0$ have a common root (when a,b,c are non-zero distinct real numbers), then which of the following is not true.

$b{x}^2+acx+b^{2}c=0andc{x}^2+abx+b^{2}c=0$

Eliminating $x^2,$ we get $x=\alpha =a$ which is  common root

Put $x=a$ in first equation

$a^{2}b+a^{2}c+b^{2}c=0\to \left(i\right)$

On dividing (i)  by $a^2{b}^2{c}^2$ we get

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

On dividing (i) by abc

$\frac{a}{c}+\frac{a}{b}+\frac{b}{c}=0$