<p>The lines <span class="math">\frac{x+3}{-3} = \frac{y-1}{1} = \frac{z-5}{5}\</span> and <span class="math">\frac{x+1}{-1} = \frac{y-2}{2} = \frac{z-5}{5}\</span> are</p>
Step-by-Step Solution
Key Concept: Two lines are coplanar if and only if the scalar triple product of the vector joining any point on one line to a point on the other line with the direction vectors of both lines equals zero.
Solution: Given: x_1 = -3, y_1 = 1, z_1 = 5 and a_1 = -3, b_1 = 1, c_1 = 5 x_2 = -1, y_2 = 2, z_2 = 5 and a_2 = -1, b_2 = 2, c_2 = 5 Consider the determinant: \begin{vmatrix} x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = \begin{vmatrix} 2 & 1 & 0 \\ -3 & 1 & 5 \\ -1 & 2 & 5 \end{vmatrix} = 0 Since the determinant equals 0, the lines are coplanar. ∴ Answer is (c)
Correct Answer: C