Question

The parametric equations of the line passing through \( A(3, 4, -7) \), \( B(1, -1, 6) \) are

• A. \[ x = 3 - 2\lambda, \quad y = 4 - 5\lambda, \quad z = -7 + 13\lambda \]
• B. \[ x = -2 + 5\lambda, \quad y = -5 + 4\lambda, \quad z = 13 - 7\lambda \]
• C. \[ x = 1 + 3\lambda, \quad y = -1 + 4\lambda, \quad z = 6 - 7\lambda \]
• D. \[ x = 3 + \lambda, \quad y = -1 + 4\lambda, \quad z = -7 + 6\lambda \]

Hint:

The parametric equations of a line passing through two points are given by the coordinates of one point plus \( \lambda \) times the direction ratios of the line.

Answer 1

The Correct Option is A

Step 1: Find the direction ratios.
The direction ratios of the line are given by the difference between the coordinates of points \( A \) and \( B \): \[ \text{Direction ratios} = (1 - 3, -1 - 4, 6 - (-7)) = (-2, -5, 13). \]
Step 2: Write the parametric equations.
The parametric equations of the line are given by: \[ x = x_1 + \lambda a, \quad y = y_1 + \lambda b, \quad z = z_1 + \lambda c, \] where \( (x_1, y_1, z_1) \) is the point \( A(3, 4, -7) \), and \( (a, b, c) \) is the direction ratio vector \( (-2, -5, 13) \). Thus, the parametric equations of the line are: \[ x = 3 - 2\lambda, \quad y = 4 - 5\lambda, \quad z = -7 + 13\lambda. \]
Step 3: Conclusion.
The parametric equations are \( x = 3 - 2\lambda \), \( y = 4 - 5\lambda \), and \( z = -7 + 13\lambda \), which corresponds to option (A).