91Ó°ÊÓ

In coordinate geometry, a parabola is a curve where any given point is equidistant from a fixed point known as the focus and a line called the directrix. Parabolas can open upwards, downwards, to the right, or to the left, depending on their orientation in the coordinate system. The standard equation of a parabola opening right is given by \( y^2 = 4ax \), where \(a\) is the distance from the vertex to the focus. In our problem, we have two parabolas with equations \( y^2 = 4ax \) and \( y^2 = 4bx \), both opening to the right. These equations categorize how the parabolas stretch along the x-axis, determined by the parameters \(a\) and \(b\). When the values of \(a\) and \(b\) are different, the parabolas exhibit different widths and positions relative to their shared axis.

A locus is the set of points satisfying certain conditions or meeting specific criteria in plane geometry. In this context, when we talk about the locus of points, we're describing a path or area that satisfies the conditions of the midpoints between the intersections of two parabolas, given by the lines parallel to the common axis. The locus found here is the collection of all possible midpoints \((h, k)\). To find the locus, we first solve for the x-coordinates \(x_1\) and \(x_2\) where the parabolas intersect the line \(y=c\) and then determine the midpoint. This results in the path being defined by an equation \[ h = \frac{k^2}{8} \left( \frac{1}{a} + \frac{1}{b} \right) \], depending on the constants \(a\) and \(b\). The shape and type of this locus can vary depending on whether \(a\) equals or differs from \(b\), turning it into different geometric figures.

The latus rectum of a parabola is a line segment that passes through the focus and is perpendicular to the axis of symmetry of the parabola. For a parabola described by \( y^2 = 4ax \), its latus rectum has a length of \(4a\). This measurement gives insight into the width of the parabola at its focus point and plays a critical role in defining its geometry. By considering the parabolas \( y^2 = 4ax \) and \( y^2 = 4bx \) in this context, the latus rectum aids in narrowing down the structure and form of the parabolas based on \(a\) and \(b\), specifying how 'wide' or 'narrow' they might appear. Understanding this concept is crucial when analyzing the intersection of parabolas and the resulting midpoints that form their locus.

The midpoint of a segment is the average position of its endpoints, providing a central location along the line. In this problem, calculating midpoints involves the intercepts on lines parallel to the axis of parabolas. Once we determine where a line \(y=c\) crosses our parabolas, the x-coordinates \(x_1\) and \(x_2\) can be found. The midpoint of these intersection points \((x_1, c)\) and \((x_2, c)\) is crucial to determining the locus. We use the formula \( \left( \frac{x_1 + x_2}{2}, c \right) \) to identify this position. Through this, the locus of midpoints gives us a general expression showing the relationship between \(h\), \(k\), and the parameters \(a\) and \(b\). This formula unveils a path or shape formed by all possible midpoints, driven by the varying values of \(c\).