Question

If the perpendicular distance of the plane passing through the point \(Q(1,0,-1)\) and containing the line \(\overset{⃗}{r} = (\hat{i}-3\hat{j}+\hat{k})+λ(2\hat{i}-2\hat{j}+\hat{k})\) from origin is \(\frac{p}{\sqrt{53}}\) then \(p =\) ...

• A. \(4\)
• B. \(1\)
• C. \(5\)
• D. \(9\)

Hint:

Find the normal as the cross product of the line direction and a vector joining a point on the line to Q.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The plane contains the line with point \(P(1,-3,1)\) and direction \(\vec d=(2,-2,1)\), and also passes through \(Q(1,0,-1)\).

Step 2: Normal vector
\[ \overrightarrow{PQ}=(0,3,-2) \]
\[ \vec n=\vec d\times\overrightarrow{PQ}=\begin{vmatrix}\hat i&\hat j&\hat k\\2&-2&1\\0&3&-2\end{vmatrix}=(1,4,6) \]

Step 3: Equation of the plane
Through \(Q(1,0,-1)\):
\[ 1(x-1)+4y+6(z+1)=0\Rightarrow x+4y+6z+5=0 \]

Step 4: Distance from the origin
\[ \frac{|5|}{\sqrt{1+16+36}}=\frac{5}{\sqrt{53}} \]
Comparing with \(\dfrac{p}{\sqrt{53}}\) gives \(p=5\), option (C).

Final Answer:
The plane is x + 4y + 6z + 5 = 0, which is 5 over root 53 from the origin, so p = 5, option (C). \[ \boxed{5} \]