Ellipse
Tangent to Ellipse
Grade 11
Question:
<p>(A) The minimum area of triangle formed by the tangent to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) and coordinate axes is:</p>
<p>(a) \(ab\) sq. units</p>
<p>(b) \(\frac{a^2 + b^2}{2}\) sq. units</p>
<p>(c) \(\frac{(a+b)^2}{2}\) sq. units</p>
<p>(d) \(\frac{a^2 + ab + b^2}{3}\) sq. units</p>
Step-by-Step Solution
Key Concept: Use parametric form of ellipse and minimize area using calculus
<p>The tangent to the ellipse at point \((a\cos\theta, b\sin\theta)\) is \(\frac{x\cos\theta}{a} + \frac{y\sin\theta}{b} = 1\). The intercepts on axes are \(\frac{a}{\cos\theta}\) and \(\frac{b}{\sin\theta}\). Area of triangle \(= \frac{1}{2} \cdot \frac{a}{\cos\theta} \cdot \frac{b}{\sin\theta} = \frac{ab}{2\sin\theta\cos\theta}\). Minimum occurs when \(\sin\theta\cos\theta\) is maximum, i.e., when \(\sin\theta = \cos\theta = \frac{1}{\sqrt{2}}\). Minimum area \(= ab\) sq. units.</p>
Correct Answer: a