If \(y_{1}, y_{2}\) and \(y_{3}\) are the ordinates of the vertices of a triangle inscribed in the parabola \(y^{2}=4 a x\), then its area is (a) \(\left|(1 / 8 \mathrm{a})\left(\mathrm{y}_{1}-\mathrm{y}_{2}\right)\left(\mathrm{y}_{2}-\mathrm{y}_{3}\right)\left(\mathrm{y}_{3}-\mathrm{y}_{1}\right)\right|\) (b) \(\left|(1 / 4 \mathrm{a})\left(\mathrm{y}_{1}-\mathrm{y}_{2}\right)\left(\mathrm{y}_{2}-\mathrm{y}_{3}\right)\left(\mathrm{y}_{3}-\mathrm{y}_{1}\right)\right|\) (c) \(\left|(1 / 2 \mathrm{a})\left(\mathrm{y}_{1}-\mathrm{y}_{2}\right)\left(\mathrm{y}_{2}-\mathrm{y}_{3}\right)\left(\mathrm{y}_{3}-\mathrm{y}_{1}\right)\right|\) (a) \(\left|(1 / a)\left(y_{1}-y_{2}\right)\left(y_{2}-y_{3}\right)\left(y_{3}-y_{1}\right)\right|\)

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves the study of geometric figures through the use of a coordinate system. This fusion of algebra and geometry allows us to represent and analyze shapes via equations and numerical coordinates. For instance, the equation of a parabola might look daunting at first glance, but it's actually a straightforward way to describe its shape.

In the context of the given exercise, we use coordinate geometry to pinpoint the location of vertices of a triangle inscribed within a parabola with the equation \(y^2 = 4ax\). Each vertex's position is defined by an x-coordinate and a y-coordinate. These coordinates are not random; they are derived systematically by substituting the y-coordinate into the parabola’s equation, thus determining the corresponding x-coordinate.

Understanding how to work with these coordinates is crucial for solving problems involving shapes on a plane, such as finding the area of a triangle inscribed in a curve, which requires precise determination of the vertices through their coordinates.

Parabolas are unique figures in geometry, represented by a U-shaped curve that possesses several noteworthy properties. They are defined as the locus of points equidistant from a fixed point called the 'focus' and a line called the 'directrix'. The standard form of the parabola equation used in our problem is \(y^2 = 4ax\), which opens to the right with the vertex at the origin.

Several key properties come into play when inscribing triangles within parabolas. For instance, the axis of symmetry of the parabola, which is the line that splits it into two mirror-image halves, helps us find symmetrical points, and the reflective property of parabolas affects the path of lines intersecting it.

In our exercise, the knowledge about parabolas helps us understand how the ordinates, the y-coordinates, of our inscribed triangle's vertices relate to the parabola, and this relationship is essential for calculating the triangle’s area.

The cross product is a powerful tool in vector algebra, particularly in three-dimensional space, used to find a vector perpendicular to two given vectors. However, when we're confined to two dimensions, as we are with a triangle on the coordinate plane, the cross product's magnitude (which would be the length of that perpendicular vector in 3D) actually gives us twice the area of the parallelogram formed by the two vectors.

When we shift our focus to triangles, we note that a vector represents each side of the triangle. The area of a triangle is simply half the area of the parallelogram that would be formed by two of its sides. Thus, calculating the cross product of two sides of the triangle and then taking half of that gives us the area of the triangle.

In the problem at hand, the vectors \(\bar{AB}\) and \(\bar{AC}\) are used to compute the cross product, and subsequently, the area of the triangle inscribed within the parabola. This cross product method provides a quick and precise way to find the area without the need for traditional, more time-consuming geometric methods.