In each of the Exercises 7 to 15, find the equations of the hyperbola satisfying the given conditions: Foci (± 4, 0), the latus rectum is of length 12
x²/4 - y²/12 = 1
Explanation
This question requires finding the equation of a hyperbola given the foci and latus rectum length. The solution follows the same pattern as Example 16 in the textbook, where a similar problem is solved with foci on the y-axis. Here, since foci are on the x-axis, the standard form x²/a² - y²/b² = 1 is used.
Solution Steps
Step 1: Since foci are (±4, 0), the hyperbola has foci on x-axis. Therefore, c = 4 and the equation is of the form x²/a² - y²/b² = 1.
Step 2: Length of latus rectum = 2b²/a = 12, which gives b² = 6a.
Step 3: For hyperbola, c² = a² + b². Substituting c = 4 and b² = 6a: 16 = a² + 6a.
Step 4: Solving a² + 6a - 16 = 0: a = [-6 ± ]/2 = [-6 ± 10]/2. Since a > 0, a = 2.
Step 5: Therefore, b² = 6a = 6(2) = 12.
Step 6: The equation of the hyperbola is x²/4 - y²/12 = 1.