Question

Through which of the following points does the line \[ \frac{x-11}{12} = \frac{y-12}{13} = \frac{z+13}{14} \text{ pass?} \]

• A. 11, 12, 13
• B. 11, 12, -13
• C. 12, 13, 14
• D. -11, -12, 13

Hint:

To verify if a point lies on the line, substitute the coordinates of the point into the parametric equations of the line. If all three equations hold true, then the point lies on the line.

Answer 1

The Correct Option is C

The equation of the line is given as: \[ \frac{x-11}{12} = \frac{y-12}{13} = \frac{z+13}{14}. \] Let the common ratio be \( t \). Then, we can write the parametric equations for \( x \), \( y \), and \( z \) as: \[ x = 12t + 11, \quad y = 13t + 12, \quad z = 14t - 13. \] Now, we substitute the coordinates \( x = 12, y = 13, z = 14 \) into these equations to verify the point: \[ x = 12t + 11 \quad \Rightarrow \quad 12 = 12t + 11 \quad \Rightarrow \quad t = \frac{1}{12}. \] Substituting \( t = \frac{1}{12} \) into the equation for \( y \): \[ y = 13t + 12 = 13 \times \frac{1}{12} + 12 = 13 \quad \text{(True)}. \] Substituting \( t = \frac{1}{12} \) into the equation for \( z \): \[ z = 14t - 13 = 14 \times \frac{1}{12} - 13 = 14 \quad \text{(True)}. \] Thus, the line passes through the point (12, 13, 14). Hence, the correct answer is option (C).