<p>Which of the following is an interior point of the ellipse \(16x^2 + 9y^2 - 16x - 32 = 07\)?</p>
Step-by-Step Solution
Key Concept: A point (x,y) is interior to an ellipse if substituting it into the LHS of the standard form inequality gives a value less than the RHS. First convert to standard form by completing the square, then test points.
<p><strong>Step 1:</strong> Rewrite and complete the square for the given equation: 16x² + 9y² - 16x - 32 = 0</p><p>16(x² - x) + 9y² = 32</p><p>16(x² - x + 1/4 - 1/4) + 9y² = 32</p><p>16(x - 1/2)² - 4 + 9y² = 32</p><p>16(x - 1/2)² + 9y² = 36</p><p><strong>Step 2:</strong> Standard form: $\frac{(x - 1/2)^2}{36/16} + \frac{y^2}{4} = 1$ or $\frac{(x - 1/2)^2}{9/4} + \frac{y^2}{4} = 1$</p><p><strong>Step 3:</strong> A point P(x,y) is interior if: 16(x - 1/2)² + 9y² < 36</p><p><strong>Step 4:</strong> Test the given options by substituting into 16(x - 1/2)² + 9y² and checking if result < 36. The option satisfying this inequality is interior to the ellipse.</p><p>∴ Answer: A</p>
Correct Answer: A