Let O P → = α − 1 α i ^ + j ^ + k ^ , O Q → = i ^ + β − 1 β j ^ + k ^ \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} OP = α α − 1 i ^ + j ^ + k ^ , OQ = i ^ + β β − 1 j ^ + k ^ and O R → = i ^ + j ^ + 1 2 k ^ \overrightarrow{O R}=\hat{i}+\hat{j}+\frac{1}{2} \hat{k} OR = i ^ + j ^ + 2 1 k ^ be three vectors, where α , β ∈ R − { 0 } \alpha, \beta \in \mathbb{R}-\{0\} α , β ∈ R − { 0 } and O O O denotes the origin. If ( O P → × O Q → ) ⋅ O R → = 0 (\overrightarrow{O P} \times \overrightarrow{O Q}) \cdot \overrightarrow{O R}=0 ( OP × OQ ) ⋅ OR = 0 and the point ( α , β , 2 ) (\alpha, \beta, 2) ( α , β , 2 ) lies on the plane 3 x + 3 y − z + l = 0 3 x+3 y-z+l=0 3 x + 3 y − z + l = 0 , then the value of l l l is ____________.
Step-by-Step Verified Solution
a m p ; ( O P → × O Q → ) ⋅ O R → = 0 a m p ; ∣ α − 1 α a m p ; 1 a m p ; 1 1 a m p ; β − 1 β a m p ; 1 1 a m p ; 1 a m p ; 1 2 ∣ = 0 \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} am p ; ( OP × OQ ) ⋅ OR = 0 am p ; α α − 1 1 1 am p ; 1 am p ; β β − 1 am p ; 1 am p ; 1 am p ; 1 am p ; 2 1 = 0
α + β + 1 = 0 ... (i) Also ( α , β , 2 ) lies on 3 x + 3 y − z + l = 0 ⇒ 3 α + 3 β − 2 + l = 0 ⇒ l = 2 − 3 ( α + β ) use (1) in it ⇒ l = 5 \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} α + β + 1 = 0 ... (i) Also ( α , β , 2 ) lies on 3 x + 3 y − z + l = 0 ⇒ 3 α + 3 β − 2 + l = 0 ⇒ l = 2 − 3 ( α + β ) use (1) in it ⇒ l = 5