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Question 1199602: Find the equation of the ellipse with center at (0,0), vertices at (2,0)
and (-2,0), and eccentricity of 2/7. Sketch the ellipse.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! **1. Determine the value of 'a'**
* The vertices are at (±2, 0), which indicates that the major axis lies along the x-axis.
* The distance from the center (0, 0) to a vertex is the length of the semi-major axis (a).
* Therefore, a = 2
**2. Determine the value of 'c'**
* Eccentricity (e) = c/a
* where:
* e = eccentricity (2/7)
* c = distance from the center to a focus
* a = semi-major axis (2)
* 2/7 = c / 2
* c = 4/7
**3. Determine the value of 'b'**
* The relationship between a, b, and c in an ellipse is:
* c² = a² - b²
* (4/7)² = 2² - b²
* 16/49 = 4 - b²
* b² = 4 - 16/49
* b² = 176/49
* b = √(176/49) = (4√11) / 7
**4. Write the equation of the ellipse**
* The standard equation of an ellipse centered at the origin with the major axis along the x-axis is:
* x²/a² + y²/b² = 1
* Substitute the values of a² and b²:
* x²/2² + y²/[(4√11)/7]² = 1
* x²/4 + y²/(176/49) = 1
* x²/4 + 49y²/176 = 1
**Therefore, the equation of the ellipse is x²/4 + 49y²/176 = 1**
**Sketching the Ellipse**
1. **Plot the center:** (0, 0)
2. **Plot the vertices:** (2, 0) and (-2, 0)
3. **Determine the endpoints of the minor axis:**
* Since b = (4√11)/7, the endpoints of the minor axis are (0, (4√11)/7) and (0, -(4√11)/7)
4. **Sketch the ellipse:**
* Draw a smooth curve connecting the vertices and the endpoints of the minor axis.
**Key Points:**
* The major axis of the ellipse lies along the x-axis.
* The eccentricity (2/7) indicates that the ellipse is relatively elongated.
I hope this helps! Let me know if you have any other questions.
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