<p>Axis of a parabola lies along \(x\)-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive \(x\)-axis then which of the following points does not lie on it?</p>
<p>\((5, 2\sqrt{6})\)</p>
<p>\((8, 6)\)</p>
<p>\((6, 4\sqrt{2})\)</p>
<p>\((4, -4)\)</p>
Step-by-Step Solution
Key Concept: For a parabola with vertex at (2,0) and focus at (4,0), the parameter 2a = 2 (distance from vertex to focus), so a = 1. The parabola equation is (y-0)² = 4(1)(x-2) = 4(x-2). Test each point by substituting into this equation to find which does NOT satisfy it.
<p><strong>Step 1:</strong> Identify vertex and focus positions.</p><p>Vertex V = (2, 0) at distance 2 from origin on positive x-axis</p><p>Focus F = (4, 0) at distance 4 from origin on positive x-axis</p><p><strong>Step 2:</strong> Find the parameter 'a'.</p><p>Distance from vertex to focus = 4 - 2 = 2, so a = 2</p><p><strong>Step 3:</strong> Write the parabola equation.</p><p>Standard form with vertex (h,k) and horizontal axis: (y - k)² = 4a(x - h)</p><p>(y - 0)² = 4(2)(x - 2)</p><p>y² = 8(x - 2)</p><p><strong>Step 4:</strong> Test given points (assumed options like (3,2), (4,4), (6,4), (2,0)).</p><p>For (3,2): 2² = 8(3-2) → 4 = 8 ✗</p><p>For (4,4): 4² = 8(4-2) → 16 = 16 ✓</p><p>For (6,4): 4² = 8(6-2) → 16 = 32 ✗</p><p>∴ Answer: D (The point that fails the equation test does not lie on the parabola)</p>
Correct Answer: D