3D Geometry
Line and Plane
Grade 12
Question:
<p>The equation of a line of greatest slope can be</p>
<p>(a) \(\frac{x}{3} = \frac{y}{1} = \frac{z}{-1}\)</p>
<p>(b) \(\frac{x}{3} = \frac{y}{-1} = \frac{z}{1}\)</p>
<p>(c) \(\frac{x}{1} = \frac{y}{-3} = \frac{z}{-1}\)</p>
<p>(d) \(\frac{x}{1} = \frac{y}{3} = \frac{z}{-1}\)</p>
Step-by-Step Solution
Key Concept: Use the direction ratios found from the previous problem and the fact that the line passes through a known point.
Step 1: From the previous problem, the direction cosines are \(\frac{3}{11}, \frac{-1}{11}, \frac{1}{11}\), which means the direction ratios are \((3, -1, 1)\). Step 2: The line of greatest slope passes through the origin (since the plane \(2x + y - 5z = 0\) passes through the origin). Step 3: The equation is \(\frac{x}{3} = \frac{y}{1} = \frac{z}{-1}\). ∴ Answer is (a).
Correct Answer: A