If y1, y2, y3 are the ordinates of the vertices of a...

If $\mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{y}_{3}$ are the ordinates of the vertices of a triangle inscribed in the parabola $\mathrm{y}^{2}=4 \mathrm{ax}(\mathrm{a}>0)$, then its area is (a) $\frac{1}{8 \mathrm{a}}\left|\mathrm{y}_{1} \mathrm{y}_{2} \mathrm{y}_{3}\right|$ (b) $\frac{1}{8 \mathrm{a}}\left|\left(\mathrm{y}_{1}-\mathrm{y}_{2}\right)\left(\mathrm{y}_{2}-\mathrm{y}_{3}\right)\left(\mathrm{y}_{3}-\mathrm{y}_{1}\right)\right|$ (c) $\frac{1}{8 \mathrm{a}}\left|\left(\mathrm{y}_{1}+\mathrm{y}_{2}+\mathrm{y}_{3}\right)\right|$ (d) $8 a\left|\left(y_{1}-y_{2}\right)\left(y_{2}-y_{3}\right)\left(y_{3}-y_{1}\right)\right|$

If $\mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{y}_{3}$ are the ordinates of the vertices of a triangle inscribed in the parabola $\mathrm{y}^{2}=4 \mathrm{ax}(\mathrm{a}>0)$, then its area is
(a) $\frac{1}{8 \mathrm{a}}\left|\mathrm{y}_{1} \mathrm{y}_{2} \mathrm{y}_{3}\right|$
(b) $\frac{1}{8 \mathrm{a}}\left|\left(\mathrm{y}_{1}-\mathrm{y}_{2}\right)\left(\mathrm{y}_{2}-\mathrm{y}_{3}\right)\left(\mathrm{y}_{3}-\mathrm{y}_{1}\right)\right|$
(c) $\frac{1}{8 \mathrm{a}}\left|\left(\mathrm{y}_{1}+\mathrm{y}_{2}+\mathrm{y}_{3}\right)\right|$
(d) $8 a\left|\left(y_{1}-y_{2}\right)\left(y_{2}-y_{3}\right)\left(y_{3}-y_{1}\right)\right|$

Key Concepts

Expressing Area in Terms of Coordinate Differences

When the vertices of a triangle are expressed through parameters or derived from a curve's properties, the area formula often simplifies to an expression involving the differences between these parameters. This symmetry, typically seen in factors such as (y? - y?), (y? - y?), and (y? - y?), reflects the invariant nature of the area under translation and permutation of the vertices, and is useful for identifying concise forms of the area in terms of given ordinates.

Determinant Method for Computing Triangle Area

The area of a triangle with vertices given by coordinates can be efficiently calculated using the determinant formula. By setting up a matrix with the coordinates of the vertices, the area is found as one half the absolute value of the determinant. This method is especially powerful in analytic geometry because it systematically uses the coordinate differences to yield an algebraic expression representing the area.

Coordinate Transformation Using the Parabola Equation

In problems where vertices of figures lie on a parabola, the inherent relation between the x- and y-coordinates (for example, x = y²/(4a) in the case of y² = 4ax) allows one to express the coordinates entirely in terms of one variable. This transformation is key to reducing the problem to a single variable form when computing geometric quantities like areas, streamlining the analysis by eliminating extra unknowns.

Parameterization of Curves

Parameterization is a technique used to represent points on a curve using a single parameter. For the parabola y² = 4ax, one can express the coordinates of any point on the curve in terms of its ordinate or another parameter. This method simplifies the process of analyzing geometric properties and calculating measures, notably when the problem provides one coordinate (the ordinate) and the other coordinate must be derived from the curve's equation.

Parabola as a Conic Section

A parabola is a type of conic section defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). In analytic geometry, its standard equations, like y² = 4ax for a > 0, play a central role in studying its geometric properties and relationships with other figures, such as triangles inscribed within it.

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