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 determine the equation of a hyperbola with the given information, we can use the standard form of a hyperbola:
(x - h)² / a² - (y - k)² / b² = 1,
where (h, k) represents the coordinates of the center, a represents the distance from the center to the vertex, and b represents the distance from the center to the foci.
Step 1: Find the center of the hyperbola.
The given vertex is (6, 5). In the standard form, the vertex is (h, k). Therefore, the center of the hyperbola is also (6, 5).
Step 2: Find the value of a.
The distance from the center to the vertex is a.
Step 3: Find the equation of the asymptotes.
The equation of the asymptotes is given by 5x - 6y - 30 = 0 and 5x + 6y - 30 = 0.
Step 4: Determine the value of b.
The distance from the center to the foci is b.
Now, let's solve it step by step:
Step 1: The center of the hyperbola is (6, 5).
Step 2: To find the value of a, we can use the distance formula between the center and the vertex:
a = distance = √[(x₂ - x₁)² + (y₂ - y₁)²],
where (x₁, y₁) = center of the hyperbola = (6, 5),
(x₂, y₂) = vertex = (6, 5).
Plugging in the values, we get:
a = √[(6 - 6)² + (5 - 5)²]
= √[0 + 0]
= √0
= 0.
Therefore, the value of a is 0.
Step 3: The equation of the asymptotes is given as 5x - 6y - 30 = 0 and 5x + 6y - 30 = 0. Rewriting both equations in slope-intercept form:
5x - 6y - 30 = 0
-6y = -5x + 30
y = (5/6)x - 5
5x + 6y - 30 = 0
6y = -5x + 30
y = (-5/6)x + 5.
The slopes of the asymptotes are 5/6 and -5/6.
Step 4: To determine the value of b, we can use the relation between a, b, and the slopes of the asymptotes.
The equation for the slopes of the asymptotes is given as b/a = ± (slope).
Plugging the values, we get:
b/0 = ± (5/6).
Since a cannot be zero, b is undefined.
Therefore, the given information does not provide enough data to determine the equation of the hyperbola.