In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions: Ends of major axis (0, ± ), ends of minor axis (± 1, 0)
The equation of the ellipse is x² + y²/5 = 1. Since the ends of the major axis are (0, ±), the major axis lies along the y-axis. Therefore, the equation of the ellipse is of the form x²/b² + y²/a² = 1. From the given conditions:
- Ends of major axis: (0, ±) gives a = , so a² = 5
- Ends of minor axis: (±1, 0) gives b = 1, so b² = 1
Substituting these values in the standard equation: x²/1 + y²/5 = 1
Therefore, the required equation is x² + y²/5 = 1.
Explanation
This question tests understanding of ellipse orientation and standard form. When the major axis ends have x-coordinate 0, the major axis is along the y-axis. The standard form becomes x²/b² + y²/a² = 1 where a > b. The semi-major axis length 'a' equals the y-coordinate of the major axis endpoint, and semi-minor axis 'b' equals the x-coordinate of the minor axis endpoint.
Solution Steps
Step 1: Identify that major axis ends (0, ±) indicate major axis is along y-axis
Step 2: Determine a = from major axis endpoints, so a² = 5
Step 3: Determine b = 1 from minor axis endpoints (±1, 0), so b² = 1
Step 4: Write equation in form x²/b² + y²/a² = 1, substituting values to get x² + y²/5 = 1