Question 1190881: Write the equation of the ELLIPSE that satisfies the given conditions
b. Vertices (-1.3) and (5, 3) length of minor axis 4
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to find the equation of the ellipse:
1. **Find the Center:** The center of the ellipse is the midpoint of the line segment connecting the vertices. The midpoint formula is ((x₁ + x₂)/2, (y₁ + y₂)/2).
Center = ((-1 + 5)/2, (3 + 3)/2) = (2, 3)
2. **Determine the Orientation:** Since the vertices have the same y-coordinate, the major axis is horizontal.
3. **Find 'a' (Semi-major Axis):** The semi-major axis is the distance from the center to a vertex.
a = distance from (2, 3) to (5, 3) = |5 - 2| = 3
4. **Find 'b' (Semi-minor Axis):** The length of the minor axis is given as 4. The semi-minor axis is half of this length.
b = 4 / 2 = 2
5. **Write the Equation:** The general equation of an ellipse centered at (h, k) with a horizontal major axis is:
(x - h)² / a² + (y - k)² / b² = 1
Substitute the values we found (h = 2, k = 3, a = 3, b = 2):
(x - 2)² / 3² + (y - 3)² / 2² = 1
Simplify:
(x - 2)² / 9 + (y - 3)² / 4 = 1
Therefore, the equation of the ellipse is (x - 2)²/9 + (y - 3)²/4 = 1.
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