Find the foci of the following hyperbola. x^2/36 - y^2/225=1
First, we need to identify the coefficients a and b:
a = √(225) = 15
b = √(36) = 6
Then, we can use the formula to find the distance from the center to each focus:
c = √(a^2 + b^2)
c = √(225 + 36)
c = √261
Therefore, the foci are located at (±c, 0):
(-√261, 0) and (√261, 0)
To find the foci of a hyperbola, we need to determine the center, the vertices, and the value of c.
First, rewrite the equation in standard form by dividing both sides by 225:
(x^2/36) - (y^2/225) = 1
The equation is now in the form:
(x^2/a^2) - (y^2/b^2) = 1
Comparing this to the standard form, we have:
a^2 = 36
b^2 = 225
Next, we can find the value of c using the relationship c^2 = a^2 + b^2. Substitute the values of a and b into the equation:
c^2 = 36 + 225
c^2 = 261
Taking the square root of both sides, we find:
c = √261
Now we have all the information needed to find the foci.
The center of the hyperbola is (0, 0).
The value of c is √261.
The distance between the center and the vertices is a = 6. (a = √36)
To find the foci, we need to add or subtract c to/from the x-coordinate of the center.
Therefore, the foci are located at (-√261, 0) and (√261, 0).
To find the foci of a hyperbola, you need to determine the center, the vertices, and the value of "c" of the hyperbola equation.
The standard equation for a hyperbola with horizontal transverse axis is given by:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1,
where (h,k) represents the center of the hyperbola. Comparing this with the provided equation, we can rewrite it as:
(x - 0)^2 / 6^2 - (y - 0)^2 / 15^2 = 1.
From this equation, we can determine that the center is (0, 0), and the values of a and b are 6 and 15, respectively.
The formula for finding the value of c is c = √(a^2 + b^2). Substituting the values of a and b, we get:
c = √(6^2 + 15^2) = √(36 + 225) = √261.
Now, the foci lie on the transverse axis, which is horizontal. Since the center is (0, 0), the foci will be located at (±c, 0).
Therefore, the foci of the given hyperbola are (√261, 0) and (-√261, 0).