In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions: Length of minor axis 16, foci (0, ± 6).
The equation of the ellipse is x²/64 + y²/100 = 1.
Explanation
This question tests understanding of ellipse properties when the major axis is along the y-axis. From the foci coordinates (0, ±6), we identify that the major axis lies on the y-axis, so the standard form x²/b² + y²/a² = 1 applies. The minor axis length gives b = 8, and using the relation c² = a² - b² with c = 6, we find a² = 100.
Solution Steps
Step 1: Since foci are at (0, ±6), the major axis is along the y-axis. The equation takes the form x²/b² + y²/a² = 1.
Step 2: Length of minor axis = 16, so semi-minor axis b = 16/2 = 8, giving b² = 64.
Step 3: Foci are (0, ±c), so c = 6. Using the relation c² = a² - b², we get 36 = a² - 64.
Step 4: Solving, a² = 100. Therefore, the equation of the ellipse is x²/64 + y²/100 = 1.