Write an equation of a hyperbola with the given foci and vertices. foci (+-17,0), vertices (+-8,0)
The center of the hyperbola is the midpoint between the vertices, so the center is (0,0).
The distance from the center to each vertex is a=8.
The distance from the center to each focus is c=17.
The equation for a hyperbola is:
((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1
where (h,k) is the center of the hyperbola and a, b are the distances from the center to the vertices and foci, respectively.
Using these values, we can plug them into the equation to get:
((x - 0)^2 / 8^2) - ((y - 0)^2 / b^2) = 1
Simplifying, we get:
(x^2 / 64) - (y^2 / b^2) = 1
To find b, we can use the relationship between a, b, and c in a hyperbola:
c^2 = a^2 + b^2
Substituting in the values we know, we get:
17^2 = 8^2 + b^2
289 = 64 + b^2
b^2 = 225
b = ±15
Therefore, the equation of the hyperbola is:
(x^2 / 64) - (y^2 / 225) = 1.
To write the equation of a hyperbola with the given foci and vertices, we need to determine the values of a, b, and h.
In a hyperbola, the distance from the center to either vertex is represented by 'a', and the distance from the center to either focus is represented by 'c'. The value of 'b' can be found using the formula: b^2 = c^2 - a^2.
Given that the foci are at (+-17,0) and the vertices are at (+-8,0), we can determine the values of 'a' and 'c' as follows:
a = 8 (since the distance from the center to any vertex is 8)
c = 17 (since the distance from the center to any focus is 17)
Next, let's calculate the value of 'b':
b^2 = c^2 - a^2
b^2 = 17^2 - 8^2
b^2 = 289 - 64
b^2 = 225
b = 15
Now we know the values of 'a', 'b', and 'c' needed to write the equation of the hyperbola. The standard equation for a hyperbola with a horizontal transverse axis is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Since the center of the hyperbola is (h, k), we need to find the values of 'h' and 'k' using the given information.
The center is the midpoint between the vertices of the hyperbola, which are (+-8,0). Thus, the center is (0, 0).
Substituting the values of 'a', 'b', 'h', and 'k' into the equation, we get:
(x - 0)^2 / 8^2 - (y - 0)^2 / 15^2 = 1
Simplifying, we have:
x^2 / 64 - y^2 / 225 = 1
Therefore, the equation of the hyperbola with the given foci (+-17,0) and vertices (+-8,0) is:
x^2 / 64 - y^2 / 225 = 1