A conic section is a curve on a plane that is defined by a \(2^\text\)-degree polynomial equation in two variables. Conic sections are classified into four groups: parabolas, circles, ellipses, and hyperbolas. Conic sections received their name because they can each be represented by a cross section of a plane cutting through a cone.
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The practical applications of conic sections are numerous and varied. They are used in physics, orbital mechanics, and optics, among others. In addition to this, each conic section is a locus of points, a set of points that satisfies a condition. Their status as loci of points allows them to be used in practical problems in which the location of an object can vary, but it needs to meet certain conditions. Understanding the coordinate geometry of conic sections allows one to model these situations with the equations of conic sections.
Each conic section can be defined as a locus of points. A locus of points is a set of points, each location of which is satisfied by some condition. These definitions are important because they inform how to use conic sections in real-world problems.
The following sections are meant to be references. Feel free to check the linked main articles for more in-depth examples and problems.
Parabola
Main Article: Equation of a Parabola
A parabola is defined in terms of a line, called the directrix, and a point not on the directrix, called the focus. A parabola is the locus of points that are equidistant from both the directrix and focus.
The axis of symmetry is the line which divides the parabola symmetrically. The vertex of the parabola is the intersection of the axis of symmetry and the parabola.
A parabola which opens vertically has the equation
\[(x-h)^2=4p(y-k).\]
\((h,k)\) are the the coordinates of the vertex. The directrix is defined by the equation \(y=k-p\). The focus has coordinates \((h,k+p)\).
A parabola which opens horizontally has the equation
\[(y-k)^2=4p(x-h).\]
\((h,k)\) are the coordinates of the vertex. The directrix is defined by the equation \(x=h-p\). The focus has coordinates \((h+p,k)\).
Circle
Main Article: Equation of a Circle
A circle is defined in terms of a point, called the center, and a non-zero length, called the radius. A circle is the locus of points located a radius away from the center:
\[(x-h)^2+(y-k)^2=r^2.\]
\((h,k)\) are the coordinates of the center of the circle and \(r\) is the radius of the circle.
Ellipse
Main Article: Equation of an Ellipse
An ellipse is defined in terms of two points, called foci (plural of focus). An ellipse is the locus of points for which the sum of the distances to each of the foci is a constant amount. This constant amount is equal to the length of the major axis:
\[\frac+\frac=1.\]
\((h,k)\) are the coordinates of the center of the ellipse. The center is the midpoint of the two foci. The chord which passes through the two foci is called the major axis. The chord that is perpendicular to the major axis and passes through the center is called the minor axis.
If \(a>b\), then the ellipse will have a horizontal major axis of length \(2a\) and a vertical minor axis of length \(2b\). The foci will be located at \(\left(h-\sqrt,k\right)\) and \(\left(h+\sqrt,k\right)\).
If \(a\right)\) and \(\left(h,k+\sqrt\right)\).
If \(a=b\), then the ellipse is a circle. A circle is sometimes considered to be a sub-type of ellipses in which the two foci coincide with one another.
\[\dfrac+\dfrac=1\] \[(x-4)^2+y^2=25\] \[y^2=-4(x-9)\] \[\dfrac-y^2=1\]Hyperbola
Main Article: Equation of a Hyperbola
A hyperbola is defined in terms of two points, called foci. A hyperbola is the locus of points for which the absolute difference of the distances to each of the foci is a constant amount. This constant amount is equal to the distance between the vertices of the hyperbola. The foci are located on a line called the transverse axis. The midpoint of the two foci is the center.
The standard form equation of a hyperbola with a horizontal transverse axis is
\[\frac-\frac=1.\]
The vertices are located at \((h-a,k)\) and \((h+a,k)\). The foci are located at \(\left(h-\sqrt,k\right)\) and \(\left(h+\sqrt,k\right)\).
The standard form equation of a hyperbola with a vertical transverse axis is
\[\frac-\frac=1.\]
The vertices are located at \((h,k-a)\) and \((h,k+a)\). The foci are located at \(\left(h,k-\sqrt\right)\) and \(\left(h,k+\sqrt\right)\).
In either case, \((h,k)\) are the coordinates of the center of the hyperbola. One can draw a rectangle centered around this point with side lengths \(2a\) and \(2b\). The lines containing the diagonals of this rectangle are the asymptotes of the hyperbola.
Brent is planning his running route. He would like the route to meet the following conditions:
