Analytic Geometry

In addition to lines and the figures that are made with them, algebraic equations exist for other types of geometric figures. One of the most common examples is the circle. A circle is defined as a figure created by the set of all points in a plane that are a constant distance from a central point. If the center of the circle is at the origin, the formula for the circle is r² = x² + y² where r is the distance of each point from the center and called the radius. For example, if a radius of 4 is chosen, a plot of all the x and y pairs that satisfy the equation 4 ² = x² + y² would create a circle. Note, this equation, which is similar to the distance formula, is called the center-radius form of the equation. When the radius of the circle is at the point (a,b) the formula, known as the general form, becomes r² = (x — a)² + (y — b)².

The circle is one kind of a broader type of geometric figures known as conic sections. Conic sections are formed by the intersection of a geometric plane and a double-napped cone. After the circle, the most common conics are parabolas, ellipses, and hyperbolas.

Curves known as parabolas are found all around us. For example, they are the shape formed by the sagging of telephone wires or the path a ball travels when it is thrown in the air. Mathematically, these figures are described as a curve created by the set of all points in a plane at a constant distance from a fixed point, known as the focus, and a fixed line, called the directrix. This means that if we take any point on the parabola, the distance of the point from the focus is the same as the distance from the directrix. A line can be drawn through the focus perpendicular to the directrix. This line is called the axis of symmetry of the parabola. The midpoint between the focus and the directrix is the vertex.

The equation for a parabola is derived from the distance formula. Consider a parabola that has a vertex at point (h,k). The linear equation for the directrix can be represented by y = k — p, where p is the distance of the focus from the vertex. The standard form of the equation of the parabola is then (x — h)² = 4p(y — k). In this case, the axis of symmetry is a vertical line. In the case of a horizontal axis of symmetry, the equation becomes (y — k)² = 4p(x — h) where the equation for the directrix is x = h — p. This formula can be expanded to give the more common quadratic formula which is y = Ax² + Bx + C, such that A does not equal 0.

Another widely used conic is an ellipse, which looks like a flattened circle. An ellipse is formed by the graph of the set of points, the sum of whose distances from two fixed points (foci) is constant. To visualize this definition, imagine two tacks placed at the foci. A string is knotted into a circle and placed around the two tacks. The string is pulled taut with a pencil and an ellipse is formed by the path traced. Certain parts of the ellipse are given various names. The two points on an ellipse intersected by a line passing through both foci are called the vertices. The chord connecting both vertices is the major axis and the chord perpendicular to it is known as the minor axis. The point at which the chords meet is known as the center.

Again by using the distance formula, the equation for an ellipse can be derived. If the center of the ellipse is at point (h,k) and the major and minor axes have lengths of 2a and 2b respectively, the standard equation is

If the center of the ellipse is at the origin, the equation simplifies to (x²/a²) + (y²/b²) = 1.

The "flatness" of an ellipse depends on a number called the eccentricity. This number is given by the ratio of the distance from the center to the focus divided by the distance from the center to the vertex. The greater the eccentricity value, the flatter the ellipse.

Another conic section is a hyperbola, which looks like two facing parabolas. Mathematically, it is similar in definition to an ellipse. It is formed by the graph of the set of points, the difference of whose distances from two fixed points (foci) is constant. Notice that in the case of a hyperbola, the difference between the two distances from fixed points is plotted and not the sum of this value as was done with the ellipse.

As with other conics, the hyperbola has various characteristics. It has vertices, the points at which a line passing through the foci intersects the graph, and a center. The line segment connecting the two vertices is called the transverse axis. The simplified equation for a hyperbola with its center at the origin is (x²/a²) — (y²/b²) = 1. In this case, a is the distance between the center and a vertex, b is the difference of the distance between the focus and the center and the vertex and the center.