a hyperbola is centered at (3,7). The vertices are (9,7) and (-3,7). The slopes of the aymptotes are m==/-5/6. Enter the equation of the hyperbola in the form: (x-h)^2/a^2-(y-k)^2/b^2=1
To find the equation of a hyperbola in the form (x-h)^2/a^2 - (y-k)^2/b^2 = 1, we need to determine the coordinates of the center (h, k), the values of a and b, and the signs of the terms (x-h)^2 and (y-k)^2.
Given that the hyperbola is centered at (3,7), we know that the center of the hyperbola is (h, k) = (3, 7).
The distance between the center of the hyperbola and the vertices represents the value of 'a'. In this case, the vertices are (9, 7) and (-3, 7). Thus, the value of 'a' is the distance between (3, 7) and (9, 7), which is 6 units. Therefore, a = 6.
The slopes of the asymptotes represent the ratio b/a. In this case, the slopes are m = ±5/6. Since the ratio b/a is the reciprocal of this slope ratio, we can determine that b/a = 6/5.
To find the value of 'b', we can multiply a by b/a:
b = a * (b/a) = 6 * (6/5) = 36/5.
Now we have all the required information to write the equation of the hyperbola:
(x-h)^2/a^2 - (y-k)^2/b^2 = 1
Substituting the known values, we get:
(x-3)^2/6^2 - (y-7)^2/(36/5)^2 = 1
Simplifying further:
(x-3)^2/36 - (y-7)^2/(1296/25) = 1
To express the equation in terms of fractions, we can simplify it as:
(x-3)^2/36 - (y-7)^2/(5184/25) = 1
Therefore, the equation of the hyperbola in the given form is:
(x-3)^2/36 - (y-7)^2/(5184/25) = 1.