Question:
<p>If the tangent at a point on the ellipse <span class="math-tex">\(\frac{x^{2}}{27}+\frac{y^{2}}{3}=1\)</span> meets the coordinate axes at A and B, and O is the origin, then the minimum area (in sq. units) of the triangle OAB is:</p>
<p style="display:inline"><span class="math-tex">\(3 \sqrt{3}\)</span></p>
<p style="display:inline">9</p>
<p style="display:inline"><span class="math-tex">\(\frac{9}{\sqrt{3}}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{9}{2}\)</span></p>
Step-by-Step Solution
Key Concept: Use the parametric form of the tangent to determine the axial intercepts and minimize the area by maximizing the sine function in the denominator.
<p>Equation of a tangent to the ellipse<br />
̣<span class="math-tex">$\frac{x}{\sqrt{27}} \cos \theta+\frac{y}{\sqrt{3}} \sin \theta=1$</span><br />
The area bounded by line and co-ordinate axis<br />
<span class="math-tex">$\Delta=\frac{1}{2}, \frac{\sqrt{27}}{\cos \theta} \cdot \frac{\sqrt{3}}{\sin \theta}=\frac{9}{\sin 2 \theta}$</span><br />
<span class="math-tex">$\Delta$</span>= will be minimum when sin 2<span class="math-tex">$\theta$</span> = 1<br />
<span class="math-tex">$\Delta_{\min }$</span> = 9</p>
Correct Answer: B