In each of the Exercises 1 to 6, find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas: x²/16 - y²/9 = 1
For the hyperbola x²/16 - y²/9 = 1:
Coordinates of Foci: (±5, 0)
Coordinates of Vertices: (±4, 0)
Eccentricity: 5/4
Length of Latus Rectum: 9/2
Explanation
This question is from Exercise 10.4, Q1. The student needs to compare the given equation with the standard form x²/a² - y²/b² = 1. From the context, Example 14 shows the methodology: identify a² and b², then calculate c² = a² + b² (note: for hyperbola, c² = a² + b², unlike ellipse). The foci are at (±c, 0), vertices at (±a, 0), eccentricity e = c/a, and latus rectum = 2b²/a.
Solution Steps
Step 1: Compare x²/16 - y²/9 = 1 with standard form x²/a² - y²/b² = 1. We get a² = 16, so a = 4; and b² = 9, so b = 3.
Step 2: Calculate c² = a² + b² = 16 + 9 = 25, so c = 5.
Step 3: Foci are at (±c, 0) = (±5, 0). Vertices are at (±a, 0) = (±4, 0).
Step 4: Eccentricity e = c/a = 5/4.
Step 5: Length of latus rectum = 2b²/a = 2(9)/4 = 9/2.