Question

A perpendicular is drawn from the point P(2, 4, -1) to the line \(\frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9}\). The equation of the perpendicular from P to the given line is

• A. \(\frac{x - 2}{6} = \frac{y - 4}{3} = \frac{z + 1}{2}\)
• B. \(\frac{x + 2}{6} = \frac{y - 4}{3} = \frac{z + 1}{2}\)
• C. \(\frac{x + 2}{-6} = \frac{y - 4}{3} = \frac{z + 1}{2}\)
• D. \(\frac{x + 2}{6} = \frac{y + 4}{3} = \frac{z + 1}{2}\)

Hint:

Foot of perpendicular is obtained by projecting point onto line.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Find foot of perpendicular, then equation of line through P and foot.
Step 2: Detailed Explanation:
Line direction ratios: \((1, 4, -9)\)
Let foot \(N = (-5 + t, -3 + 4t, 6 - 9t)\)
\(\vec{PN} = (-7 + t, -7 + 4t, 7 - 9t)\)
\(\vec{PN} \cdot (1, 4, -9) = 0\)
\((-7 + t) + 4(-7 + 4t) - 9(7 - 9t) = 0\)
\(-7 + t - 28 + 16t - 63 + 81t = 0\)
\(-98 + 98t = 0 \rightarrow t = 1\)
\(N = (-4, 1, -3)\)
\(\vec{PN} = (-6, -3, -2)\) or \((6, 3, 2)\)
Equation: \(\frac{x - 2}{6} = \frac{y - 4}{3} = \frac{z + 1}{2}\)
Step 3: Final Answer:
\(\frac{x - 2}{6} = \frac{y - 4}{3} = \frac{z + 1}{2}\).