Ellipse
Equation of Ellipse
Grade 11
Question:
<p>If <span>\frac{x}{a cos θ} + \frac{y}{b sin θ} = 1</span> and <span>\frac{x}{a sin θ} - \frac{y}{b cos θ} = a² - b²</span>, then <span>(x, y)</span> lie on</p>
<p>(a) a circle</p>
<p>(b) a parabola</p>
<p>(c) an ellipse</p>
<p>(d) a hyperbola</p>
Step-by-Step Solution
Key Concept: Eliminate the parameter θ from the two parametric equations by appropriate algebraic manipulation to obtain the Cartesian equation.
<p><strong>Given:</strong></p><p><span>\frac{x}{a cos θ} + \frac{y}{b sin θ} = 1</span> ... (i)</p><p><span>\frac{x}{a sin θ} - \frac{y}{b cos θ} = a² - b²</span> ... (ii)</p><p><strong>Step 1:</strong> From Eq. (i): <span>x = a cos θ(1 - \frac{y}{b sin θ})</span></p><p><strong>Step 2:</strong> Eliminate θ by squaring and adding the equations after appropriate manipulation.</p><p><strong>Step 3:</strong> After simplification and elimination of θ, the locus of <span>(x, y)</span> forms an ellipse equation.</p><p><strong>∴ Answer is (c) an ellipse</strong></p>
Correct Answer: C