To solve this problem, we need to determine the equation of a line that is perpendicular from point \( P(2, 4, -1) \) to the given line. The given line is represented in symmetric form as:
\[ \frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9} \]The direction ratios (d.r's) of this line are \(1, 4, -9\).
Step 1: Find the direction ratios of the perpendicular line
Let the direction ratios of the line perpendicular from point \( P(2, 4, -1) \) be \( l, m, n \). Since it is perpendicular to the given line, the dot product of their direction ratios must be zero:
\[ 1 \cdot l + 4 \cdot m + (-9) \cdot n = 0 \]This simplifies to:
\[ l + 4m - 9n = 0 \quad \text{(Equation 1)} \]Step 2: Find the direction ratios using given options
We will test each option to see which one satisfies the condition from Equation 1.
Option 1: \(\frac{x - 2}{6} = \frac{y - 4}{3} = \frac{z + 1}{2}\)
Direction ratios: \(6, 3, 2\)
\[ 1 \cdot 6 + 4 \cdot 3 + (-9) \cdot 2 = 6 + 12 - 18 = 0 \]This satisfies Equation 1.
Option 2: \(\frac{x + 2}{6} = \frac{y - 4}{3} = \frac{z + 1}{2}\)
Direction ratios: \(6, 3, 2\)
\[ 1 \cdot 6 + 4 \cdot 3 + (-9) \cdot 2 = 6 + 12 - 18 = 0 \]This also satisfies Equation 1, but does not use the point \( P(2, 4, -1) \).
Option 3: \(\frac{x + 2}{-6} = \frac{y - 4}{3} = \frac{z + 1}{2}\)
Direction ratios: \(-6, 3, 2\)
\[ 1 \cdot (-6) + 4 \cdot 3 + (-9) \cdot 2 = -6 + 12 - 18 \neq 0 \]This does not satisfy Equation 1.
Option 4: \(\frac{x + 2}{6} = \frac{y + 4}{3} = \frac{z + 1}{2}\)
Direction ratios: \(6, 3, 2\)
\[ 1 \cdot 6 + 4 \cdot 3 + (-9) \cdot 2 = 6 + 12 - 18 = 0 \]This satisfies Equation 1, but the orientation with respect to \( P \) is not considered.
Conclusion: The correct option that satisfies the condition and uses the point correctly is: