If a, b, c and vmatrix b c a c a b vmatrix=0, then (a)/( -a)+(b)/( -b)+( )/( -c) is equal …

Question

If $\alpha 
e a,\beta 
e b,\gamma 
e c$ and $\begin{vmatrix} \alpha  \quad  b \quad  c \ a \quad  \beta  \quad  c \ a \quad  b \quad  \gamma \end{vmatrix}=0$, then $\frac{a}{\alpha -a}+\frac{b}{\beta -b}+\frac{\gamma }{\gamma -c}$ is equal to :

Accepted Answer

$$\left|\[\begin{array}{lll} \alpha & b & c \ a & \beta & c \ a & b & \gamma \end{array}$$\right|=0\]

$$\begin{aligned}
& R_1 \rightarrow R_1-R_2, R_2 \rightarrow R_2-R_3 \
& \Rightarrow\left|\[\begin{array}{ccc} \alpha-a & b-\beta & 0 \ 0 & \beta-b & c-\gamma \ a & b & \gamma \end{array}\end{aligned}$$\right|=0
\]

Take $$\alpha$$-a, $$\beta$$-b, $$\gamma$$-c common from column-1, 2 and 3 respectively

$$\begin{aligned}
& (\alpha-a)(\beta-b)(\gamma-c)\left|\[\begin{array}{ccc} 1 & -1 & 0 \ 0 & 1 & -1 \ \frac{a}{\alpha-a} & \frac{b}{\beta-b} & \frac{\gamma}{\gamma-c} \end{array}\end{aligned}$$\right|=0 \
\& \Rightarrow \frac{\gamma}{\gamma-c}+\frac{b}{\beta-b}+\frac{a}{\alpha-a}=0
\]

Solutions · 1