Question

The sum of the coordinates of one of the points on the line \(\frac{x-2}{1} = \frac{y+3}{-2} = \frac{z+5}{2}\) which is at a distance of 3 units from the point \((2,-3,-5)\) is ...

• A. \(7\)
• B. \(-7\)
• C. \(11\)
• D. \(-11\)

Hint:

Use the determinant of the joining vector and the two direction vectors.

Answer 1

The Correct Option is B

Step 1: Points and directions:
Line 1 passes through \(P(-1, 3, 1)\) with direction \(\vec d_1 = (1, -\lambda, \lambda)\). Line 2 passes through \(Q(0, 1, 2)\) with direction \(\vec d_2 = \left(\frac12, 1, -1\right)\), proportional to \((1, 2, -2)\).

Step 2: Coplanarity condition:
\(\overrightarrow{PQ} = (1, -2, 1)\). The lines are coplanar if
\[ \begin{vmatrix}1 & -2 & 1\\1 & -\lambda & \lambda\\1 & 2 & -2\end{vmatrix} = 0 \]

Step 3: Expand:
\[ 1(2\lambda - 2\lambda) + 2(-2 - \lambda) + 1(2 + \lambda) = 0 - 4 - 2\lambda + 2 + \lambda = -2 - \lambda \]
So \(-2 - \lambda = 0\), giving \(\lambda = -2\).

Final Answer:
\(\lambda = -2\), option (C). \[ \boxed{-2} \]