A) \[\mathbf{r}.(\lambda \mathbf{a}-\mu \mathbf{b})=0\]
B) \[\mathbf{r}.\,(\lambda \mathbf{b}-\mu \mathbf{a})=0\]
C) \[\mathbf{r}.(\lambda \mathbf{a}+\mu \mathbf{b})=0\]
D) \[\mathbf{r}.(\lambda \mathbf{b}+\mu \mathbf{a})=0\]
Correct Answer: B
Solution :
The equation of a plane through the line of intersection of the planes \[\mathbf{r}.\mathbf{a}=\lambda \] and \[\mathbf{r}.\,\mathbf{b}=\mu \] can be written as \[(\mathbf{r}.\mathbf{a}-\lambda )+k(\mathbf{r}.\mathbf{b}-\mu )=0\] or \[\mathbf{r}.(\mathbf{a}+k\mathbf{b})=\lambda +k\mu \] .....(i) This passes through the origin, therefore \[\mathbf{0}.(\mathbf{a}+k\mathbf{b})=\lambda +\mu k\Rightarrow k=\frac{-\lambda }{\mu }\] Putting the value of k in (i), we get the equation of the required plane as \[\mathbf{r}.(\mu \mathbf{a}-\lambda \mathbf{b})=0\ \Rightarrow \ \ \mathbf{r}\ .\ (\lambda \mathbf{b}-\mu \mathbf{a})=0\].
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