MHT CET · Maths · Three Dimensional Geometry
Let \(P(3,2,6)\) be a point in space and \(Q\) be a point on the line \(\bar{r}=\hat{i}-\hat{j}+2 \hat{k}+\mu(-3 \hat{i}+\hat{j}+5 \hat{k})\). Then the value of \(\mu\) for which the vector \(\overline{\mathrm{PQ}}\) is parallel to the plane \(x-4 y+3 z=1\) is
- A \(\frac{1}{4}\)
- B \(-\frac{1}{4}\)
- C \(\frac{1}{8}\)
- D \(-\frac{1}{8}\)
Answer & Solution
Correct Answer
(A) \(\frac{1}{4}\)
Step-by-step Solution
Detailed explanation
Let the position vector of Q be
\(\begin{array}{ll}
& (\hat{i}-\hat{j}+2 \hat{k})+\mu(-3 \hat{i}+\hat{j}+5 \hat{k}) \\
& =(-3 \mu+1) \hat{i}+(\mu-1) \hat{j}+(5 \mu+2) \hat{k} \\
\therefore \quad & \overline{P Q}=(-3 \mu-2) \hat{i}+(\mu-3) \hat{j}+(5 \mu-4) \hat{k}
\end{array}\)
Since \(\overline{\mathrm{PQ}}\) is parallel to the plane,
\(\begin{aligned}
& (-3 \mu-2)(1)+(\mu-3)(-4)+(5 \mu-4)(3)=0 \\
& \Rightarrow \mu=\frac{1}{4}
\end{aligned}\)
\(\begin{array}{ll}
& (\hat{i}-\hat{j}+2 \hat{k})+\mu(-3 \hat{i}+\hat{j}+5 \hat{k}) \\
& =(-3 \mu+1) \hat{i}+(\mu-1) \hat{j}+(5 \mu+2) \hat{k} \\
\therefore \quad & \overline{P Q}=(-3 \mu-2) \hat{i}+(\mu-3) \hat{j}+(5 \mu-4) \hat{k}
\end{array}\)
Since \(\overline{\mathrm{PQ}}\) is parallel to the plane,
\(\begin{aligned}
& (-3 \mu-2)(1)+(\mu-3)(-4)+(5 \mu-4)(3)=0 \\
& \Rightarrow \mu=\frac{1}{4}
\end{aligned}\)
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