If the points A(1, 2, 3), B(3, p, 7) and C(q, −1, 9) are collinear, then the value of p + q is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 4
The points are collinear when \(\overrightarrow{AC}\) is a scalar multiple of \(\overrightarrow{AB}\):
\(\overrightarrow{AB} = 2\hat{\imath}+(p-2)\hat{\jmath}+4\hat{k}\)
\(\overrightarrow{AC} = (q-1)\hat{\imath}-3\hat{\jmath}+6\hat{k}\)
The z-components fix the multiple:
\( \displaystyle \lambda = \frac{6}{4} = \frac{3}{2} \)
So \(\overrightarrow{AC} = \frac{3}{2}\overrightarrow{AB}\). Comparing the other components:
\( \displaystyle q-1 = \frac{3}{2}\times 2 = 3 \), so \(q = 4\)
\( \displaystyle -3 = \frac{3}{2}(p-2) \), so \(p-2 = -2\) and \(p = 0\)
Hence
\(p+q = 0+4 = 4\)