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Let's check each point to see if it lies on the line \(x=1+5\mu\), \(y=-5+\mu\), \(z=-6-3\mu\).
(A) \((1, -5, 6)\)
If this point lies on the line, there must be a value of \(\mu\) that satisfies all three equations:
\(1 = 1 + 5\mu \Rightarrow 5\mu = 0 \Rightarrow \mu = 0\)
\(-5 = -5 + \mu \Rightarrow \mu = 0\)
\(6 = -6 - 3\mu \Rightarrow 12 = -3\mu \Rightarrow \mu = -4\)
Since the values of \(\mu\) are not consistent, point (A) does not lie on the line.
(B) \((1, 5, 6)\)
\(1 = 1 + 5\mu \Rightarrow \mu = 0\)
\(5 = -5 + \mu \Rightarrow \mu = 10\)
\(6 = -6 - 3\mu \Rightarrow 12 = -3\mu \Rightarrow \mu = -4\)
Since the values of \(\mu\) are not consistent, point (B) does not lie on the line.
(C) \((1, -5, -6)\)
\(1 = 1 + 5\mu \Rightarrow \mu = 0\)
\(-5 = -5 + \mu \Rightarrow \mu = 0\)
\(-6 = -6 - 3\mu \Rightarrow -3\mu = 0 \Rightarrow \mu = 0\)
Since \(\mu = 0\) satisfies all three equations, point (C) lies on the line.
(D) \((-1, -5, 6)\)
\(-1 = 1 + 5\mu \Rightarrow 5\mu = -2 \Rightarrow \mu = -\frac{2}{5}\)
\(-5 = -5 + \mu \Rightarrow \mu = 0\)
\(6 = -6 - 3\mu \Rightarrow 12 = -3\mu \Rightarrow \mu = -4\)
Since the values of \(\mu\) are not consistent, point (D) does not lie on the line.
Correct Answer: (C) \((1, -5, -6)\)