find the standard form of the hyperbola, the center and vertices given the foci (0, -+8) and asymptotes at y=4x
since the asymptotes are y=4x, the center must be at (0,0)
The foci are on the y-axis, so the asymptotes are y = ±a/b x
since c=8 (from the foci)
and a/b = 4,
b^2 + (4b)^2 = 8^2
b = 8/√17
a = 32/√17
y^2/a^2 - x^2/b^2 = 1
y^2/(1024/17) - x^2/(64/17) = 1
or
17y^2/1024 - 17x^2/64 = 1
see the graph at
http://www.wolframalpha.com/input/?i=hyperbola+y%5E2%2F(1024%2F17)+-+x%5E2%2F(64%2F17)+%3D+1
To find the standard form of a hyperbola, the center, and vertices, given the foci and the equations of the asymptotes, we can follow these steps:
Step 1: Identify the center of the hyperbola:
The center of the hyperbola is the midpoint between the two foci. In this case, the foci are at (0, -8) and (0, 8). The midpoint formula is used to find the center:
Center = ((x₁ + x₂)/2, (y₁ + y₂)/2)
= ((0 + 0)/2, (-8 + 8)/2)
= (0, 0)
So, the center of the hyperbola is at (0, 0).
Step 2: Find the distance between the center and the foci:
The distance between the center and each focus is called "c." In this case, c = 8 (distance between the center and one of the foci).
Step 3: Determine the distance between the center and the vertices:
The distance between the center and each vertex is called "a." To find "a," we need to use the given equation of the asymptote.
In this case, the equation of the asymptote is y = 4x.
To obtain the distance between the center and the vertices, we need to find the slope of the asymptote by comparing it to the standard form of a hyperbola.
The standard form of the equation of a hyperbola is:
(x-h)²/a² - (y-k)²/b² = 1
The slope of the asymptote, 4 in this case, is equal to b/a. We can set b = 1 (since the slope of the asymptote is 4) and solve for a:
4 = 1/a
a = 1/4
So, the distance between the center and the vertices is 1/4.
Step 4: Determine the value of "b":
Using the relationship between a, b, and c, we can find the value of b:
c² = a² + b²
8² = (1/4)² + b²
64 = 1/16 + b²
63/16 = b²
b = sqrt(63/16)
Step 5: Write the equation in standard form:
Now we have all the information needed to write the equation in standard form:
(x-h)²/a² - (y-k)²/b² = 1
Plugging in the values we found:
(x-0)²/(1/4)² - (y-0)²/(sqrt(63/16))² = 1
Simplifying:
x²/(1/16) - y²/(63/16) = 1
16x² - (63/16)y² = 1
Therefore, the standard form of the hyperbola is 16x² - (63/16)y² = 1.
The center of the hyperbola is (0, 0).
The distance between the center and the vertices is 1/4.
The distance between the center and the foci is 8.