91Ó°ÊÓ

The equation of a parabola is a mathematical expression that describes its shape in a coordinate plane. The standard form of a parabola is expressed as \( y = ax^2 + bx + c \). In this equation:

• \( a \) determines the direction and width of the parabola. If \( a \) is positive, the parabola opens upwards, and if negative, it opens downwards.
• \( b \) affects the position of the vertex along the x-axis.
• \( c \) is the y-intercept, showing where the parabola crosses the y-axis.

For the given problem, the equation is more complex: \( y = \frac{a^3 x^2}{3} + \frac{a^2 x}{2} - 2a \). Here, the coefficients are functions of the parameter \( a \). This makes the parabola part of a family of parabolas that vary as \( a \) changes.

The vertex formula is crucial for locating the highest or lowest point of a parabola, also known as its vertex. The x-coordinate of the vertex is calculated using the formula \( x = -\frac{B}{2A} \), derived from the general parabolic equation \( y = Ax^2 + Bx + C \).In our scenario:

• \( A = \frac{a^3}{3} \)
• \( B = \frac{a^2}{2} \)

Plug these into the vertex formula:\[ x = -\frac{\frac{a^2}{2}}{2 \times \frac{a^3}{3}} = -\frac{3}{4a} \]This formula provides the x-coordinate of the vertex, which is essential for determining the locus of the parabola’s vertices as \( a \) varies. Understanding the vertex helps in comprehending how the parabola moves across the coordinate plane.

Locus calculation involves finding a set of points that satisfy a particular condition, forming a shape or path in coordinate space. In the context of this problem, we are tasked with finding the locus of the vertices of a family of parabolas as a parameter \( a \) changes.We determine both the x and y coordinates of the vertex for the parabola equation:

• x-coordinate: \( x = -\frac{3}{4a} \)
• y-coordinate: Substitute \( x = -\frac{3}{4a} \) into the equation to find \( y = -\frac{35a}{16} \)

By examining how these coordinates relate, we calculate the product \( xy \):\[ xy = \left(-\frac{3}{4a}\right) \left(-\frac{35a}{16}\right) = \frac{105}{64} \]Thus, the locus of these vertices forms the line given by \( xy = \frac{105}{64} \). This represents a fixed relationship between the x and y coordinates of the vertices as the parameter \( a \) varies.

Coordinate geometry, also known as analytic geometry, is the study of geometric figures through an algebraic approach using a coordinate system. In this field, we can use algebraic equations to describe geometric shapes and define relationships between points and lines.For parabolas, coordinate geometry allows us to:

• Plot their paths on a Cartesian plane using an equation.
• Find intercepts, slopes, and more complex features like the focus and directrix.
• Analyze and predict how changes in equation parameters (like \( a \) in our parabola) affect the shape's graph.

This exercise centers on understanding how the vertex location changes as a function of the parameter \( a \), exemplifying coordinate geometry's power to analyze the movement and transformation of shapes. By applying coordinate geometry skills, students can visualize and predict how altering elements in an equation affects all aspects of its graph.