Question

If the points \(A(1,p,1),B(2,5,-1),C(q,11,r)\) are collinear. If the line passing through these points is perpendicular to the vector \(4\hat{i}-2\hat{j}+\hat{k}\), then the value of \(p+q+r\) is

• A. \(-4\)
• B. \(3\)
• C. \(4\)
• D. \(-1\)

Hint:

Use the direction of AB, apply the perpendicularity condition to find p, then use proportionality for BC to get q and r.

Answer 1

The Correct Option is D

Step 1: Find p First
Dot product of $\overrightarrow{AB}$ with $(4,-2,1)$ is zero: $4-2(5-p)-2=2p-8=0$, so $p=4$.

Step 2: Use the Y Coordinates
Along the line the y-coordinate rises by 1 for each unit of x. From B to C, y rises by 6.

Step 3: Find q and r
So x rises by 6 and z falls by 12: $q=2+6=8$, $r=-1-12=-13$.
Thus $p+q+r=-1$, option (D).

Final Answer:
The sum is $-1$. \[ \boxed{\text{(D) }-1} \]