3D Geometry
Lines and Circles
Grade 12
Question:
<p>The line \(\frac{x-2}{3} = \frac{y+1}{2} = \frac{z-1}{-1}\) intersects the curve \(x^2 + y^2 = r^2, z = 0\) then</p>
<p>(a) Equation of the plane through (0, 0, 0) perpendicular to the given line is \(3x + 2y - z = 0\)</p>
<p>(b) \(r = \sqrt{26}\)</p>
<p>(c) \(r = 6\)</p>
<p>(d) \(r = 7\)</p>
Step-by-Step Solution
Key Concept: Find the intersection of a line with a circle in the xy-plane by setting z=0 and solving parametrically.
Step 1: The line has direction vector \((3, 2, -1)\). Setting \(z = 0\) on the line: \(\frac{x-2}{3} = \frac{y+1}{2} = \frac{0-1}{-1} = 1\). So \(x = 5, y = 1\). Step 2: The intersection point is \((5, 1, 0)\). Since this lies on \(x^2 + y^2 = r^2\), we have \(25 + 1 = r^2\), so \(r = \sqrt{26}\).
Correct Answer: B