Parabola
Geometric Properties
Grade 11
Question:
<p>For a quadratic equation ax² + bx + c = 0 where ab = c, if the roots are α and β with roots of y = ax² + bx + c corresponding to points, show that (OT)² = (OP)(OQ) where OT = c/a.</p>
Step-by-Step Solution
Key Concept: The geometric mean relationship holds when ab = c, creating a harmonic division of the x-axis.
<p><strong>Step 1:</strong> Given ab = c from the condition on the equation.</p><p><strong>Step 2:</strong> The points P and Q have x-coordinates at a and b respectively.</p><p><strong>Step 3:</strong> Therefore: $$(OT)^2 = \left(\frac{c}{a}\right)^2 = (OP)(OQ) = ab = \frac{c}{a}$$</p><p>∴ Answer is (d).</p>
Correct Answer: D