question_answer

The equation of plane which passes through the point     of     intersection     of     lines \[\vec{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda (3\hat{i}+\hat{j}+2\hat{k})\] and \[\vec{r}=3\hat{i}+\hat{j}+2\hat{k}+\mu (\hat{i}+2\hat{j}+3\hat{k})\] where \[\lambda ,\mu \in R\] and has the greatest distance from the origin is

A) \[\vec{r}.(7\hat{i}+2\hat{j}+4\hat{k})=54\]

B) \[\vec{r}.(5\hat{i}+4\hat{j}+3\hat{k})=57\]

C) \[\vec{r}.(3\hat{i}+4\hat{j}+5\hat{k})=49\]

D) \[\vec{r}.(4\hat{i}+3\hat{j}+5\hat{k})=50\]

Correct Answer: D

Solution :

\[\therefore \] Equation of plane is \[4(x-4)+3(y-3)\,+5(z-5)=0\]\[\Rightarrow \,\vec{r}.(4\hat{i}+3\hat{j}+5\hat{k})=50\]