Q74 / 110 · VITEEE 2022 · Mathematics
The equation of the line passing through \((1,2,3)\) and parallel to the planes \(r \cdot(\hat{i}-\hat{j}+2 \hat{k})=5\) and r. \((3 \hat{i}+\hat{j}+\hat{k})=6\) is
- A\(r=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})\)
- B\(r=(-3 \hat{i}+5 \hat{j}+4 \hat{k})+\lambda(\hat{i}+2 \hat{j}+3 \hat{k})\)
- Correct answerC\(r=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(-3 \hat{i}+5 \hat{j}+4 \hat{k})\)
- D\(r=\lambda(-3 \hat{i}+5 \hat{j}+4 \hat{k})\)
Solution
The equation of line passing through \((1,2\),
3) and parallel to \(b\) is given by
\(r=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda\left(b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\right)\) ...(i)
The equations of the given planes are
\(\begin{aligned}
& r \cdot(\hat{i}-\hat{j}+2 \hat{k})=5 \quad ...(ii)\\
& \text { and } r \cdot(3 \hat{i}+\hat{j}+\hat{k})=6 \quad ...(iii)
\end{aligned}\)
The line in eq. (i) and plane in eq. (ii) are parallel. Therefore, the normal to the plane of eq. (ii) and the given line are perpendicular.
\(\begin{aligned}
& \therefore \quad(\hat{i}-\hat{j}+2 \hat{k}) \cdot \lambda\left(b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\right)=0 \\
& \Rightarrow \quad \lambda\left(b_1-b_2+2 b_3\right)=0 \\
& \Rightarrow \quad\left(b_1-b_2+2 b_3\right)=0 \quad ...(iv)
\end{aligned}\)
Similarly,
\(\begin{aligned}
& (3 \hat{i}+\hat{j}+\hat{k}) \cdot \lambda\left(b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\right)=0 \\
& \Rightarrow \lambda\left(3 b_1+b_2+b_3\right)=0 \quad ...(v)
\end{aligned}\)
From eqs. (iv) and (v), we obtain
\(\begin{aligned}
& \frac{b_1}{(-1) \times 1-1 \times 2}=\frac{b_2}{2 \times 3-1 \times 1}=\frac{b_3}{1 \times 1-3(-1)} \\
& \Rightarrow \quad \frac{b_1}{-3}=\frac{b_2}{5}=\frac{b_3}{4}
\end{aligned}\)
Therefore, the direction ratios of \(b\) are \(-3,5\) and 4 .
\(\therefore \mathrm{b}=\mathrm{b}_1 \hat{\mathrm{i}}+\mathrm{b}_2 \hat{\mathrm{j}}+\mathrm{b}_3 \hat{\mathrm{k}}=-3 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\)
Substituting the value of \(b\) in eq. (i), we obtain
\(r=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(-3 \hat{i}+5 \hat{j}+4 \hat{k})\)
This is the equation of the required line.
Hence, the correct answer is option: 'C'.