A team of 5 persons goes on a rescue operation in a helicopter. They know the coordinates of places from where people need to be rescued. These are P(1, 1, 0), Q(1, 2, 1) and R(−2, 2, −1). They decide to stop at Point A (2, 1, 3). From A, 3 team members are dropped on ground using a rope. They reach the point B at the ground level so that distance AB is shortest from the plane passing through P, Q and R.
Based on the above information, answer the following:
What are the coordinates of point B?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\left(\displaystyle\frac{29}{11},\frac{43}{22},\frac{45}{22}\right)\)
The plane through P, Q and R is \(2x+3y-3z = 5\) (normal \(\overrightarrow{PQ}\times\overrightarrow{PR} = -2\hat{\imath}-3\hat{\jmath}+3\hat{k}\), through P(1, 1, 0)). B is the foot of the perpendicular from A(2, 1, 3), so it lies on the line through A along the normal:
\(B = (2+2t,\ 1+3t,\ 3-3t)\)
B lies on the plane:
\(2(2+2t)+3(1+3t)-3(3-3t) = 5\)
\(-2+22t = 5\)
\( \displaystyle t = \frac{7}{22} \)
Hence
\( \displaystyle B = \left(2+\frac{14}{22},\ 1+\frac{21}{22},\ 3-\frac{21}{22}\right) \)
\( \displaystyle = \left(\frac{58}{22},\frac{43}{22},\frac{45}{22}\right) = \left(\frac{29}{11},\frac{43}{22},\frac{45}{22}\right) \)
Note: This topic was not in the CUET 2026 syllabus.