The line \( x = 1 + 5\mu, y = -5 + \mu, z = -6 - 3\mu \) passes through which of the following points?
Show Hint
- \( (1, -5, 6) \)
- \( (1, 5, 6) \)
- \( (1, -5, -6) \)
- \( (-1, -5, 6) \)
The Correct Option is C
Solution and Explanation
The problem requires determining which of the provided points lies on the line defined by the parametric equations:
\( x = 1 + 5\mu \), \( y = -5 + \mu \), and \( z = -6 - 3\mu \). This is achieved by checking if a consistent value of \( \mu \) exists for each point across all three equations.
1. Evaluation of Option (A): (1, -5, 6)
Substitution into the x-equation: \( 1 = 1 + 5\mu \Rightarrow \mu = 0 \).
Checking y-coordinate: \( y = -5 + 0 = -5 \). This matches the given y-coordinate. ✔️
Checking z-coordinate: \( z = -6 - 3(0) = -6 \). This does not match the given z-coordinate of 6. ❌
Therefore, point (A) is not on the line.
2. Evaluation of Option (B): (1, 5, 6)
From the x-equation: \( 1 = 1 + 5\mu \Rightarrow \mu = 0 \).
Checking y-coordinate: \( y = -5 + 0 = -5 \). This does not match the given y-coordinate of 5. ❌
Therefore, point (B) is not on the line.
3. Evaluation of Option (C): (1, -5, -6)
From the x-equation: \( 1 = 1 + 5\mu \Rightarrow \mu = 0 \).
Checking y-coordinate: \( y = -5 + 0 = -5 \). This matches the given y-coordinate. ✔️
Checking z-coordinate: \( z = -6 - 3(0) = -6 \). This matches the given z-coordinate. ✔️
All three equations are satisfied for \( \mu = 0 \).
4. Determination:
Point (1, -5, -6) satisfies all parametric equations of the line.
Final Answer:
The correct option is (C) (1, -5, -6).
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