Did you know?
Did you know that the standard form of the equation of a hyperbola with given foci and asymptotes can easily be determined using specific steps? Let's take the example of a hyperbola with foci at (0, ±8) and asymptotes represented by the equation y = ±2x.
Step 1: Identify the center of the hyperbola. The center is the midpoint between the two foci. In this case, the center is (0,0).
Step 2: Determine the value of "a" by finding the distance between the center and one of the foci. In this case, the distance is 8 units. Therefore, a = 8.
Step 3: Find the value of "b" using the asymptotes. For a hyperbola, the ratio of "b/a" is equal to the slope of the asymptotes. The slope here is 2, so b = 2a. Substituting the value of a (8), we get b = 16.
Step 4: Now that we have the values of a and b, we can use them to write the standard equation of the hyperbola. The equation is given by:
(x^2 / a^2) - (y^2 / b^2) = 1
Substituting the values of a and b, we get:
(x^2 / 8^2) - (y^2 / 16^2) = 1
Simplifying it further will give you the standard form of the equation of the hyperbola with the given characteristics.
By following these steps, you can find the standard form of the equation of any hyperbola using its foci and asymptotes.
