If a cone is cut by a plane parallel to its axis, the intersection is a hyperbola, the only conic section made of two separate pieces, or branches. Hyperbolas occur in a number of applied settings. The navigational system called LORAN (long-range navigation) uses radio signals to locate a ship or plane at the intersection of two hyperbolas. Satellites moving with sufficient speed will follow an orbit that is branch of a hyperbola; for example, a rocket sent to the moon must be fitted with retrorockets to reduce its speed in order to achieve an elliptical, rather than hyperbolic, orbit about the moon.
If the origin is the center of the hyperbola and the foci (labeled and on the graphs above) lie on the axes, we can use the distance formula to derive its equation.
In the first case, the two branches of the hyperbola open left and right, so the graph has -intercepts at and but no -intercepts. (See figure above left.) The segment joining the -intercepts is the transverse axis, and its length is . The endpoints of the transverse axis are the vertices of the hyperbola. The segment of length is called the conjugate axis.
In the second case, the graph has -intercepts at and but no -interceptsβthe two branches open up and down. (See figure above right.) Here the -intercepts are the vertices, so the transverse axis is vertical and has length . The conjugate axis has length .
The branches of the hyperbola approach two straight lines that intersect at its center. These lines are asymptotes of the graph, and they are useful as guidelines for sketching the hyperbola. We first sketch a rectangle (called the central rectangle) whose sides are parallel to the axes and whose dimensions are and . The asymptotes are the diagonals of this rectangle.
The graph is a hyperbola with center at the origin. The -term is positive, so the branches of the hyperbola open upward and downward. Because and , we have and , and the vertices are and . There are no -intercepts.
We construct the central rectangle with dimensions and , as shown in the figure. Then we draw the asymptotes through the diagonals of the rectangle. The asymptotes have slopes . Finally, we sketch the branches of the hyperbola through the vertices and approaching the asymptotes.
The fourth conic section, after circles, ellipses, and hyperbolas, is the parabola. We have already encountered parabolas in our study of quadratic functions. In particular, the graph of has its vertex at the origin and opens up or down, depending on the sign of . The graph of is a parabola that opens to the left or right. There is also a geometric definition of a parabola, but we will not discuss that here.
The first equation describes a hyperbola whose transverse axis is parallel to the -axis, so that the branches open left and right, and the second equation describes a hyperbola whose transverse axis is parallel to the -axis, so that the branches open up and down, as shown below.
Because the -term is positive, the branches open left and right. The coordinates of the vertices are thus and , or approximately and . The ends of the conjugate axis are ) and ), or approximately and .
The central rectangle is centered at the point and extends to the vertices in the horizontal direction and to the ends of the conjugate axis in the vertical direction. We draw the asymptotes through the opposite corners of the central rectangle, and sketch the hyperbola through the vertices and approaching the asymptotes to obtain the graph shown above.
Both asymptotes pass through the center of the hyperbola, . Their slopes are
and
We substitute these values into the point-slope formula to find the equations
Thus, and , and the vertices are and . The ends of the conjugate axis are and .
The central rectangle is centered at , as shown in the figure. We draw the asymptotes through the corners of the rectangle, then sketch the hyperbola by starting at the vertices and approaching the asymptotes.
for which , the coefficient of the -term, is zero. These graphs are conic sections with axes parallel to one or both of the coordinate axes. If does not equal zero, the axes of the conic section are rotated with respect to the coordinate axes. The graphing of such equations is taken up in more advanced courses in analytic geometry.
The graph of a second-degree equation can also be a point, a line, a pair of lines, or no graph at all, depending on the values of the coefficients through . Such graphs are called degenerate conics.
we can determine the nature of the graph from the coefficients of the quadratic terms. If the graph is not a degenerate conic, the following criteria apply.
The standard forms for the conic sections are summarized in the table below. For the parabola, is the vertex of the graph, and for the other conics, is the center.
The coefficients ,, and do not figure in determining the type of conic section the equation represents. They do, however, determine the position of the graph relative to the origin. Once we recognize the form of the graph, we can write the equation in standard form in order to discover more information about the graph.
Problems 29β32 deal with the cooling tower at an electricity generating facility. The shape of the tower, called a hyperboloid, is obtained by rotating a portion of the hyperbola around the -axis.
The diameter of the tower first decreases with height and then increases again. There are two heights at which the towerβs diameter is 250 feet. Find the greater of the two heights.