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100. y 2. 25 y 2. 4 x 2. 100. 4. 100. 100. =. –. y 2 25. –. = 1. EXAMPLE 1. Graph an equation of a hyperbola. Graph 25 y 2 – 4 x 2 = 100 . Identify the vertices, foci, and asymptotes of the hyperbola. SOLUTION. STEP 1. Rewrite the equation in standard form.
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100 y2 25y2 4x2 100 4 100 100 = – y225 – = 1 EXAMPLE 1 Graph an equation of a hyperbola Graph 25y2 – 4x2 = 100. Identify the vertices, foci, and asymptotes of the hyperbola. SOLUTION STEP 1 Rewrite the equation in standard form. 25y2 – 4x2 = 100 Write original equation. Divide each side by 100. Simplify.
29. 29. soc = The foci are at( 0, + ) (0, + 5.4). 25 ab + + The asymptotes are y = x x y = or EXAMPLE 1 Graph an equation of a hyperbola STEP 2 Identify the vertices, foci, and asymptotes. Note that a2 = 4 and b2 = 25, so a = 2and b = 5. The y2 - term is positive, so the transverse axis is vertical and the vertices are at (0, +2). Find the foci. c2= a2– b2= 22– 52= 29.
EXAMPLE 1 Graph an equation of a hyperbola STEP 3 Draw the hyperbola. First draw a rectangle centered at the origin that is 2a= 4units high and 2b= 10units wide. The asymptotes pass through opposite corners of the rectangle. Then, draw the hyperbola passing through the vertices and approaching the asymptotes.
EXAMPLE 2 Write an equation of a hyperbola Write an equation of the hyperbola with foci at (– 4, 0) and (4, 0) and vertices at (– 3, 0) and (3, 0). SOLUTION The foci and vertices lie on the x-axis equidistant from the origin, so the transverse axis is horizontal and the center is the origin. The foci are each 4 units from the center, so c = 4. The vertices are each 3 units from the center, soa = 3.
= 1 x2 x2 9 32 = 1 y2 7 y2 7 – – EXAMPLE 2 Write an equation of a hyperbola Becausec2 = a2 + b2, you haveb2 = c2 – a2. Findb2. b2= c2–a2= 42–32= 7 Because the transverse axis is horizontal, the standard form of the equation is as follows: Substitute 3 for aand 7 for b2. Simplify
x2 x2 The equation is in standard form. 16 16 = 1 y2 49 y2 49 – – for Examples 1 and 2 GUIDED PRACTICE Graph the equation. Identify the vertices, foci, and asymptotes of the hyperbola. 1. = 1 SOLUTION STEP 1
65 , 0 65. soc = + The foci are at( + ) 74 + ba + The asymptotes are y = x y = or x for Examples 1 and 2 GUIDED PRACTICE STEP 2 Identify the vertices, foci, and asymptotes. Note that a2 = 16 and b2 = 49, so a = 4and b = 7. The x2 - term is positive, so the transverse axis is horizontal and the vertices are at (+4, 0). Find the foci. c2= a2+ b2= 42+ 72= 65.
for Examples 1 and 2 GUIDED PRACTICE STEP 3 Draw the hyperbola. First draw a rectangle centered at the origin that is 2a= 8units high and 2b= 14units wide. The asymptotes pass through opposite corners of the rectangle. Then, draw the hyperbola passing through the vertices and approaching the asymptotes.
The equation is in standard form. = 1 y2 y2 36 36 x2 1 – for Examples 1 and 2 GUIDED PRACTICE 2. = 1 – x2 SOLUTION STEP 1 STEP 2 Identify the vertices, foci, and asymptotes. Note that a2 = 36 and b2 = 1, so a = 6and b = 1. The y2 - term is positive, so the transverse axis is horizontal and the vertices are at (0, +6). Find the foci.
soc = + The foci are at( 0, + ) ba = + 6x + The asymptotes are y = x or 37 37 61 + y = x for Examples 1 and 2 GUIDED PRACTICE c2= a2+ b2= 62+ 12= 37.
for Examples 1 and 2 GUIDED PRACTICE STEP 3 Draw the hyperbola. First draw a rectangle centered at the origin that is 2a= 12units high and 2b= 2units wide. The asymptotes pass through opposite corners of the rectangle. Then draw the hyperbola passing through the vertices and approaching the asymptotes.
4y2 y2 9x2 9 36 36 = 1 – x24 = 1 – for Examples 1 and 2 GUIDED PRACTICE 3. 4y2 – 9x2 = 36 SOLUTION STEP 1 The equation is in standard form. 4y2 – 9x2 = 36 Write original equation. Divide each side by 36. Simplify.
The foci are at( 0, + ) soc = + 13 13 32 ba + + or The asymptotes are y = y = x x for Examples 1 and 2 GUIDED PRACTICE STEP 2 Identify the vertices, foci, and asymptotes. Note that a2 = 9 and b2 = 4, so a = 3and b = 2. The y2 - term is positive, so the transverse axis is horizontal and the vertices are at (0, +3). Find the foci. c2= a2+ b2= 32+ 22= 13.
for Examples 1 and 2 GUIDED PRACTICE STEP 3 Draw the hyperbola. First draw a rectangle centered at the origin that is 2a= 6units high and 2b= 4units wide.
4.Foci:(– 3, 0), (3, 0) Vertices:(– 1, 0), (1, 0) for Examples 1 and 2 GUIDED PRACTICE Write an equation of the hyperbola with the given foci and vertices. SOLUTION The foci and vertices lie on the x-axis equidistant from the origin, so the transverse axis is horizontal and the center is the origin. The foci are each 3 units from the center, soc = 3. The vertices are each 1 units from the center, soa = 1.
= 1 x2 y2 8 12 – = 1 x2 y2 8 – for Examples 1 and 2 GUIDED PRACTICE Becausec2 = a2 + b2, you haveb2 = c2 – a2. Findb2. b2= c2–a2= 32–12= 8 Because the transverse axis is horizontal, the standard form of the equation is as follows: Substitute 1 for aand 8 for b2. Simplify
5.Foci:(0, – 10), (0, 10) Vertices:(0, – 6), (0, 6) for Examples 1 and 2 GUIDED PRACTICE SOLUTION The foci and vertices lie on the y-axis equidistant from the origin, so the transverse axis is vertical and the center is the origin. The foci are each 10 units from the center, soc = 10. The vertices are each 6 units from the center, soa = 6. Becausec2 = a2 + b2, you haveb2 = c2 – a2. Findb2. b2= c2–a2= 102–62= 64
= 1 = 1 y2 y2 362 36 x2 64 x2 64 – – for Examples 1 and 2 GUIDED PRACTICE Because the transverse axis is horizontal, the standard form of the equation is as follows: Substitute 6 for aand 64 for b2. Simplify
