JEE Advance - Mathematics (2024 - Paper 1 Online - No. 12)
Let $\overrightarrow{O P}=\frac{\alpha-1}{\alpha} \hat{i}+\hat{j}+\hat{k}, \overrightarrow{O Q}=\hat{i}+\frac{\beta-1}{\beta} \hat{j}+\hat{k}$ and $\overrightarrow{O R}=\hat{i}+\hat{j}+\frac{1}{2} \hat{k}$ be three vectors, where $\alpha, \beta \in \mathbb{R}-\{0\}$ and $O$ denotes the origin. If $(\overrightarrow{O P} \times \overrightarrow{O Q}) \cdot \overrightarrow{O R}=0$ and the point $(\alpha, \beta, 2)$ lies on the plane $3 x+3 y-z+l=0$, then the value of $l$ is ____________.
Answer
5
Explanation
$$\begin{aligned} & (\overrightarrow{\mathrm{OP}} \times \overrightarrow{\mathrm{OQ}}) \cdot \overrightarrow{\mathrm{OR}}=0 \\ & \left|\begin{array}{ccc} \frac{\alpha-1}{\alpha} & 1 & 1 \\ 1 & \frac{\beta-1}{\beta} & 1 \\ 1 & 1 & \frac{1}{2} \end{array}\right|=0 \end{aligned}$$
$$\begin{array}{ll} \qquad\alpha+\beta+1=0 \quad \text{... (i)}\\ \text { Also } \quad (\alpha, \beta, 2) \text { lies on } 3 \mathrm{x}+3 \mathrm{y}-\mathrm{z}+l=0 \\ \Rightarrow \quad 3 \alpha+3 \beta-2+l=0 \quad \Rightarrow \quad l=2-3(\alpha+\beta) \\ \text { use (1) in it } \Rightarrow l=5 \end{array}$$
Comments (0)
