Hyperbola
Parametric Form and Extrema
Grade 11
Question:
<p>If <i>x</i>, <i>y</i> ∈ ℝ satisfy the equation <i>\[</i><span>\frac{(x+4)^2}{4}</span><i>\]</i> − <i>\[</i><span>\frac{y^2}{9}</span><i>\]</i> = 1, then the difference between the largest and smallest value of the expression <i>\[</i><span>\frac{x^2}{4}</span><i>\]</i> + <i>\[</i><span>\frac{y^2}{9}</span><i>\]</i> is</p>
<p>(A) 4</p>
<p>(B) 5</p>
<p>(C) 8</p>
<p>(D) 9</p>
Step-by-Step Solution
Key Concept: Use parametrization of the hyperbola to express the objective function in terms of a single parameter, then find its range.
<p><strong>Step 1:</strong> The constraint is a hyperbola: <i>$$</i><span>\frac{(x+4)^2}{4}</span><i>$$</i> − <i>$$</i><span>\frac{y^2}{9}</span><i>$$</i> = 1. Parametrize using <i>x</i> + 4 = 2cos<i>θ</i> and <i>y</i> = 3sin<i>θ</i>.</p><p><strong>Step 2:</strong> From the parametrization: <i>x</i> = 2cos<i>θ</i> − 4 and <i>y</i> = 3sin<i>θ</i>.</p><p><strong>Step 3:</strong> Compute <i>$$</i><span>\frac{x^2}{4}</span><i>$$</i> + <i>$$</i><span>\frac{y^2}{9}</span><i>$$</i> = <i>$$</i><span>\frac{(2\cos\theta - 4)^2}{4}</span><i>$$</i> + sin²<i>θ</i> = <i>$$</i><span>\frac{4\cos^2\theta - 16\cos\theta + 16}{4}</span><i>$$</i> + sin²<i>θ</i> = cos²<i>θ</i> − 4cos<i>θ</i> + 4 + sin²<i>θ</i> = 5 − 4cos<i>θ</i>.</p><p><strong>Step 4:</strong> Since −1 ≤ cos<i>θ</i> ≤ 1, the expression 5 − 4cos<i>θ</i> ranges from 5 − 4(1) = 1 to 5 − 4(−1) = 9.</p><p><strong>Step 5:</strong> The difference is 9 − 1 = 8.</p><p>∴ Answer is (C).</p>
Correct Answer: C