Question

The line \( x = 1 + 5\mu, y = -5 + \mu, z = -6 - 3\mu \) passes through which of the following points?

• A. \( (1, -5, 6) \)
• B. \( (1, 5, 6) \)
• C. \( (1, -5, -6) \)
• D. \( (-1, -5, 6) \)

Hint:

To find a point on a parametric line, substitute the value of the parameter into the parametric equations for \( x \), \( y \), and \( z \).

Answer 1

The Correct Option is C

To solve the problem, we are given the parametric equations of a line:
\( x = 1 + 5\mu \), \( y = -5 + \mu \), and \( z = -6 - 3\mu \)
We need to check which of the given points satisfies all three equations for some value of \( \mu \).

1. Check Option (A): (1, -5, 6)
Try substituting into the equations:
- From \( x = 1 + 5\mu \), putting \( x = 1 \):
\( 1 = 1 + 5\mu \Rightarrow \mu = 0 \)
- Check \( y = -5 + \mu = -5 + 0 = -5 \) ✔️
- Check \( z = -6 - 3\mu = -6 - 0 = -6 \) ❌ (Expected z = 6, but we get -6)
So, (A) is not on the line.

2. Check Option (B): (1, 5, 6)
- \( 1 = 1 + 5\mu \Rightarrow \mu = 0 \)
- \( y = -5 + 0 = -5 \) ❌ (Expected y = 5)
So, (B) is not on the line.

3. Check Option (C): (1, -5, -6)
- \( 1 = 1 + 5\mu \Rightarrow \mu = 0 \)
- \( y = -5 + 0 = -5 \) ✔️
- \( z = -6 - 3(0) = -6 \) ✔️
All three coordinates satisfy the line’s equations when \( \mu = 0 \).

4. Conclusion:
The point (1, -5, -6) lies on the line.

Final Answer:
The correct option is (C) (1, -5, -6).