what is the equation of a hyperbola with one vertex at (6,5) and the equation of asymptotes are 5x-6y-30 and 5x+6y-30?
To find the equation of a hyperbola given one vertex and the equations of its asymptotes, you can follow these steps:
Step 1: Identify the center of the hyperbola. Since you are given one vertex, you can consider it as one focus of the hyperbola. The center of the hyperbola lies on the midpoint of the segment connecting the given vertex to the corresponding conjugate vertex. In this case, the given vertex is (6,5).
Step 2: Find the corresponding conjugate vertex. Since the asymptotes have the form 5x - 6y - 30 and 5x + 6y - 30, the slopes of the asymptotes are ± 6/5. The conjugate axis of the hyperbola is perpendicular to the transverse axis and passes through the center. The slope of the conjugate axis is the negative reciprocal of the slope of the transverse axis, which is ± 5/6.
To find the corresponding conjugate vertex, you can move 5 units up and down from the given vertex (6,5) along the conjugate axis. Therefore, the corresponding conjugate vertices are (6, 5+5) = (6, 10) and (6, 5-5) = (6, 0).
Step 3: Determine the distance between the center and the corresponding conjugate vertices. For a hyperbola, the distance between the center and the corresponding conjugate vertices is denoted as b.
b = 10 - 5 = 5
Step 4: Identify the distance between the center and the vertices. For a hyperbola, the distance between the center and the vertices is denoted as a. In this case, since you are given one vertex, (6,5), and the center is (h, k), you can use the distance formula:
a = distance between (6,5) and (h,k)
Step 5: Write the equation of the hyperbola in the standard form. The standard form for a hyperbola with a center at (h,k) is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Substituting the values, you have:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Let's denote the coordinates of the center as (h, k). Since the given vertex is (6,5), we get:
(x - 6)^2 / a^2 - (y - 5)^2 / 5^2 = 1
Now, you need to determine the value of 'a' to complete the equation. To find 'a', you can use any of the given asymptotes. Let's take the first asymptote equation, 5x - 6y - 30:
Rearranging the equation, you get:
6y = 5x - 30
y = (5/6)x - 5
Comparing this equation with the standard form of a hyperbola, you have:
y = (a/b)x + k
From the equation, you can determine that (a/b) = 5/6, which means a = (5/6)b.
Step 6: Substitute the value of 'a' obtained above into the equation. Since a = (5/6)b, you can replace 'a' with (5/6)b, giving:
(x - 6)^2 / ((5/6)b)^2 - (y - 5)^2 / b^2 = 1
Simplifying further:
(x - 6)^2 / (25/36)b^2 - (y - 5)^2 / b^2 = 1
Finally, you have the equation of the hyperbola in terms of 'b':
36(x - 6)^2 - 25(y - 5)^2 = 25b^2