Let O be the origin. The point P on the line \(\displaystyle\frac{x-1}{1}=\frac{y-2}{-1}=\frac{z+1}{2}\) for which OP is perpendicular to the line \(\displaystyle\frac{x}{2}=\frac{y}{1}=\frac{z}{1}\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (0, 3, −3)
Let P be the general point of the first line:
\(P(1+\lambda,\ 2-\lambda,\ -1+2\lambda)\)
OP has direction ratios \(1+\lambda,\ 2-\lambda,\ -1+2\lambda\). For OP to be perpendicular to the line with direction ratios 2, 1, 1:
\(2(1+\lambda)+(2-\lambda)+(-1+2\lambda) = 0\)
\(3+3\lambda = 0\)
\(\lambda = -1\)
So P is \((1-1,\ 2+1,\ -1-2) = (0, 3, -3)\). Check: \(2(0)+3+(-3) = 0\).
The other three points are also on the first line (\(\lambda = 1, 0, -2\)), but for them \(3+3\lambda\) is 6, 3 and −3, not 0.