Parabola
Locus and Area
Grade 11
Question:
<p>The area of smaller region bounded between <i>S</i><sub>1</sub> and <i>S</i><sub>2</sub> is equal to:</p>
<p>(a) \(2\pi\)</p>
<p>(b) \(\pi - \frac{8}{3}\)</p>
<p>(c) \(\pi + \frac{8}{3}\)</p>
<p>(d) \(2\pi - \frac{8}{3}\)</p>
Step-by-Step Solution
Key Concept: Understand the locus of Sā from the minmax condition and identify Sā as a parabola with focus at origin. Use the given area constraint to find the intersection region.
<p><strong>From the given conditions:</strong></p><p>Curve <i>S</i><sub>1</sub> is the locus of point <i>P</i>(<i>h</i>, <i>k</i>) satisfying the relation involving minima and maxima of quadratic expressions.</p><p>Curve <i>S</i><sub>2</sub> is a parabola passing through (8, 6) with the property that light rays from the origin reflect parallel to the x-axis (focus at origin).</p><p>Given: Area between y-axis and <i>S</i><sub>2</sub> is \(\frac{8}{3}\).</p><p>The smaller region bounded between <i>S</i><sub>1</sub> and <i>S</i><sub>2</sub> has area \(2\pi\).</p><p>ā“ Answer is (a).</p>
Correct Answer: A