If $\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}$ and…

If $\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}$ and $\left|\begin{array}{lll}\alpha & \mathrm{b} & \mathrm{c} \\ \mathrm{a} & \beta & \mathrm{c} \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0$, then $\frac{\mathrm{a}}{\alpha-\mathrm{a}}+\frac{\mathrm{b}}{\beta-\mathrm{b}}+\frac{\gamma}{\gamma-\mathrm{c}}$ is equal to:
  1. 3
  2. 0
  3. 1
  4. 2

Solution

$\begin{aligned} & \mathrm{R}_1 \rightarrow \mathrm{R}_1-\mathrm{R}_2, \mathrm{R}_2 \rightarrow \mathrm{R}_2-\mathrm{R}_3 \\ & \left|\begin{array}{ccc}\alpha-\mathrm{a} & \mathrm{b}-\beta & 0 \\ 0 & \beta-\mathrm{b} & \mathrm{c}-\gamma \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0 \\ & (\alpha-\mathrm{a})(\gamma(\beta-\mathrm{b})-\mathrm{b}(\mathrm{c}-\gamma))-(\mathrm{b}-\beta)(-\mathrm{a}(\mathrm{c}-\gamma))=0 \\ & \gamma(\alpha-\mathrm{a})(\beta-\mathrm{b})-\mathrm{b}(\alpha-\mathrm{a})(\mathrm{c}-\gamma)+\mathrm{a}(\mathrm{b}-\beta)(\mathrm{c}-\gamma) \\ & \frac{\gamma}{\gamma-\mathrm{c}}+\frac{\mathrm{b}}{\beta-\mathrm{b}}+\frac{\mathrm{a}}{\alpha-\mathrm{a}}=0\end{aligned}$

Asked in: JEE Main 2024 (08 Apr Shift 2)

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