3 points O(0,0), P(a, a^2), Q(-b, b^2)(a>0, b>0) are on the...

3 points $O(0,0), P\left(a, a^{2}\right), Q\left(-b, b^{2}\right)(a>0, b>0)$ are on the parabola $y=x^{2} .$ Let $S_{1}$ be the area bounded by the line $P Q$ and the parabola and let $S_{2}$ be the area of the $\triangle O P Q$. the minimum value of $S_{1} / S_{2}$ is (a) $4 / 3$ (b) $5 / 3$ (c) 2 (d) $7 / 3$

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Key Concepts

Parabola Geometry

This concept involves understanding the properties of a parabola, which is the graph of a quadratic function. Recognizing that points lie on the curve y = x² is fundamental to determining their coordinates, understanding symmetry, and establishing relationships between the points on the curve.

Definite Integration for Area Calculation

This concept pertains to the use of definite integrals to calculate the area bounded by curves. In problems where a region is confined by a curve and a line, setting up appropriate integrals to represent the area under the curve and above the line is essential for finding areas between these boundaries.

Coordinate Geometry for Triangle Area

This concept involves computing the area of a triangle given its vertices in a coordinate system. Using formulas derived from determinants or geometric reasoning allows one to express the triangle’s area in terms of the coordinates of its vertices, connecting algebraic expressions to geometric figures.

Calculus Optimization

Optimization in calculus involves finding the minimum or maximum values of a function. In the context of this problem, it is used to determine the minimum value of the ratio of two areas by setting up an expression and employing differentiation to locate critical points where the optimum is attained.

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