Definite Integration
Grade 12
Question:
<p>3 points O (0, 0), P(a, a<sup>2</sup>), Q (-b, b<sup>2</sup>) (a > 0, b > 0) are on the parabola y = x<sup>2</sup>. Let S<sub>1</sub> be the area bounded by the line PQ and the parabola and let S<sub>2</sub> be the area of the triangle OPQ, the minimum value of <span class="math-tex">\(\frac{S_{1}}{S_{2}}\)</span> is :</p>
<p style="display:inline"><span class="math-tex">\(\frac{4}{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{5}{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{7}{3}\)</span></p>
<p style="display:inline">2</p>
Step-by-Step Solution
Key Concept: Express the areas $S_1$ and $S_2$ in terms of the parameters $a$ and $b$ using integration and the triangle area formula, then apply the AM-GM inequality to minimize the resulting expression.
<p><span class="math-tex">$\frac{4}{3}$</span></p>
Correct Answer: A