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Question 1186028: Find the equation of the ellipse whose center is at (2, 7), having a focus at (2, 10), and having an endpoint of its minor axis at (4, 7).
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to find the equation of the ellipse:
**1. Identify the center:**
The center of the ellipse is given as (h, k) = (2, 7).
**2. Determine the orientation:**
Since the focus (2, 10) and the center (2, 7) have the same x-coordinate, the major axis is vertical.
**3. Find the distance from the center to a focus (c):**
c = |10 - 7| = 3
**4. Find the length of the semi-minor axis (b):**
The endpoint of the minor axis is (4, 7). The distance from the center (2, 7) to this endpoint is the length of the semi-minor axis (b).
b = |4 - 2| = 2
**5. Find the length of the semi-major axis (a):**
We know that a² = b² + c². Therefore:
a² = 2² + 3²
a² = 4 + 9
a² = 13
a = √13
**6. Write the equation of the ellipse:**
The standard form equation of an ellipse with a vertical major axis is:
((x - h)² / b²) + ((y - k)² / a²) = 1
Substituting the values we found:
((x - 2)² / 2²) + ((y - 7)² / (√13)²) = 1
Simplifying:
((x - 2)² / 4) + ((y - 7)² / 13) = 1
Therefore, the equation of the ellipse is $\boxed{\frac{(x-2)^2}{4} + \frac{(y-7)^2}{13} = 1}$.
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