Question

If $x,y,z$ are different from zero and $\Delta =\left|\begin{array}{ccc}a& b-y& c-z\\ a-x& b& c-z\\ a-x& b-y& c\end{array}\right|=0$, then the value of the expression $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}$ is ?

• A. $0$
• B. $-1$
• C. $1$
• D. $2$

Answer 1

The correct option is D $2$

$\Delta =\left|\begin{array}{ccc}a& b-y& c-z\\ a-x& b& c-z\\ a-x& b-y& c\end{array}\right|=0$
$xyz\left|\begin{array}{ccc}\frac{a}{x}& \frac{b}{y}-1& \frac{c}{z}-1\\ \frac{a}{x}-1& \frac{b}{y}& \frac{c}{z}-1\\ \frac{a}{x}-1& \frac{b}{y}-1& \frac{c}{z}\end{array}\right|=0$
$\Rightarrow \left|\begin{array}{ccc}\frac{a}{x}& \frac{b}{y}-1& \frac{c}{z}-1\\ \frac{a}{x}-1& \frac{b}{y}& \frac{c}{z}-1\\ \frac{a}{x}-1& \frac{b}{y}-1& \frac{c}{z}\end{array}\right|=0$
$C_1\to C_1+C_2+C_3$
$\left|\begin{array}{ccc}\frac{a}{x}+\frac{b}{y}+\frac{c}{z}-2& \frac{b}{y}-1& \frac{c}{z}-1\\ \frac{a}{x}+\frac{b}{y}+\frac{c}{z}-2& \frac{b}{y}& \frac{c}{z}-1\\ \frac{a}{x}+\frac{b}{y}+\frac{c}{z}-2& \frac{b}{y}-1& \frac{c}{z}\end{array}\right|=0$
$(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}-2)\left|\begin{array}{ccc}1& \frac{b}{y}-1& \frac{c}{z}-1\\ 1& \frac{b}{y}& \frac{c}{z}-1\\ 1& \frac{b}{y}-1& \frac{c}{z}\end{array}\right|=0$
$R_1\to R_1-R_2,R_2\to R_2-R_3$
$(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}-2)\left|\begin{array}{ccc}0& -1& 0\\ 0& 1& -1\\ 1& \frac{b}{y}-1& \frac{c}{z}\end{array}\right|=0$
$(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}-2)=0$
$\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=2$