Hyperbolas Help (page 2)
Introduction to Hyperbolas
The last conic section is the hyperbola. Hyperbolas are formed when a slice is made through both parts of a double cone. The graph of a hyperbola comes in two pieces called branches . Like ellipses, hyperbolas have a center, two foci, and two vertices. Hyperbolas also have two slant asymptotes. The definition of a hyperbola involves the distance between points on the graph and two fixed points.
DEFINITION: A hyperbola is the set of all points such that the difference of the distance between a point and two fixed points (the foci) is constant.
For example, the foci for are (−5, 0) and (5, 0). For any point on the hyperbola, the distance between this point and one focus minus the distance between the same point and the other focus is 6. Two points on the hyperbola are .
Equations of hyperbolas come in one of two forms.
If the x 2 term is positive, one branch opens to the left and the other to the right. If the y 2 term is positive, one branch opens up and the other down. The formulas for these two forms are in Figures 12.21 and 12.22.
Sketching Hyperbolas on a Graph
We can sketch a hyperbola by plotting the vertices and sketching the asymptotes, using dashed lines. We should also plot two points to the left and two points to the right of the vertices.
- Sketch the graph for .
Because y 2 is positive, we will use the information in Figure 12.22 . The center is (0, 0), a = 2, and b = 1. The vertices are ( h , k + a ) = (0, 0 + 2) = (0, 2) and ( h , k − a ) = (0, 0 − 2) = (0, −2). The asymptote formulas are . Using our numbers for h, k, a , and b , we have y = −2 x and y = 2x . We will use x = 4 and x = −4 for our extra points. If we let x = 4 or x = −4, we get two y -values, . These give us four more . (see Figure 12.23.)
Finding the Center, Vertices, Foci, and Asymptotes of a Hyperbola
In the next problem, we will find the center, vertices, foci, and asymptotes for given hyperbolas. Once we have determined whether x 2 is positive or y 2 is positive, we can decide on which formulas to use, those in Figure 12.21 or Figure 12.22.
Find the center, vertices, foci, and asymptotes for the hyperbola.
Because x 2 is positive, we will use the information in Figure 12.21 .
h = −7, k = −4, a = 6, b = 8,
Center: (−7, −4)
Vertices: ( h − a , k ) = (−7 −6, −4) = (−13, −4) and ( h + a , k ) = (−7 + 6, −4) = (−1, −4)
Foci: ( h − c , k ) = (−7 − 10, −4) = (−17, −4) and ( h + c , k ) = (−7 + 10, −4) = (3, −4)
Because y 2 is positive, we need to use the information in Figure 12.22 .
h = 1, k = 0, a = 12, b = 5,
Center: (1, 0)
Vertices: ( h , k − a ) = (1, 0−12) = (1, −12) and ( h , k + a ) = (1, 0 + 12) = (1, 12)
Foci: ( h , k − c ) = (1, 0 − 13) = (1, −13) and ( h , k + c ) = (1, 0 + 13) = (1, 13)
Matching Hyperbola Equations with Graphs
In the next problem, we will match equations of hyperbolas with their graphs. Being able to identify the vertices will not be enough. We will also need to use the equations of the asymptotes to find b (we will know a from the vertices). Because the center of each hyperbola will be at (0, 0), the asymptotes will either be and .
Match the equation with its graph in Figures 12.24–12.27.
The vertices are (−2, 0) and (2, 0). The slopes of the asymptotes are −1 and 1. The graph is in Figure 12.25 .
The vertices are (−2, 0) and (2, 0). The slopes of the asymptotes are and . The graph is in Figure 12.26 .
The vertices are (0, −2) and (0, 2). The slopes of the asymptotes are −1 and 1. The graph is in Figure 12.27 .
The vertices are (0, −2) and (0, 2). The slopes of the asymptotes are −2 and 2. The graph is in Figure 12.24 .
Finding the Equation for a Hyperbola
We can find the equation for a hyperbola when we know some points or a point and the asymptotes. If we have the vertices and foci, then finding an equation for a hyperbola will be similar to finding an equation for an ellipse. If we are given the vertices and asymptotes or foci and asymptotes, we will need to use the slope of one of the asymptotes to find either a or b (we will know one but not the other from the vertices or foci). The first thing we need to decide is which formulas to use—those in Figures 12.21 or Figure 12.22. If the vertices or foci are on the same horizontal line (the y -coordinates are the same), we will use Figure 12.21. If they are on the same vertical line (the x -coordinates are the same), we will use Figure 12.22.
Find an equation for the hyperbola.
- The vertices are (3, −1) and (3, 7) and is an asymptote. The vertices are on the same vertical line, so we need to use the information in Figure 12.22. The vertices are ( h , k − a ) = (3, −1) and ( h , k + a ) = (3, 7). This gives us h = 3, k − a = −1 and k + a = 7.
The center is (3, 3) and a = 4. Once we have b , we will be done. The slope of one of the asymptotes in Figure 12.22 is , so we have , so b = 3. The equation is
- The vertices are (−8, 5) and (4, 5), and the foci are (−12, 5) and (8, 5). The vertices and foci are on the same horizontal line, so we need to use the information in Figure 12.21. The vertices are ( h − a , k ) = (−8, 5) and ( h + a , k ) = (4, 5). Now we know k = 5 and we have the system h − a = −8 and h + a = 4.
−2 − a = −8 Let h = −2 in h − a = − 8
a = 6
A focus is ( h − c , k ) = (−2 − c , 5) = (−12, 5), which gives us −2 − c = –12. Now that we see that c = 10, we can put this and a = 6 in to find b .
The equation is
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