How to Find the Equation of an Ellipse Given Foci and Major Axis Length
When dealing with the equation of an ellipse, understanding the given parameters like foci and the length of the major axis can help you derive the required equation. This article will guide you through the process using a specific example.
Step 1: Identify the Center and Orientation of the Ellipse
The foci of the ellipse are given as (0, -8) and (0, 8). The center of the ellipse is the midpoint of the line segment joining the foci.
Center:
Center (left(frac{0 - 8 0 8}{2}, frac{0 8 0 - 8}{2}right)) (0, 0)
Since the foci have the same x-coordinate, the ellipse is vertically oriented.
Step 2: Determine the Semi-Major Axis Length
The length of the major axis is given as 34 units. Therefore, the semi-major axis a is half the length of the major axis.
Semi-Major Axis Length:
a (frac{34}{2}) 17
Step 3: Determine the Distance c from the Center to the Foci
The distance c from the center to each focus is the distance from the center (0, 0) to either focus (0, 8) or (0, -8).
Distance c:
c 8
Step 4: Calculate the Semi-Minor Axis Length b
The relationship between a, b, and c in an ellipse is given by c^2 a^2 - b^2.
Calculation:
8^2 17^2 - b^2
64 289 - b^2
b^2 289 - 64 225
b (sqrt{225}) 15
Step 5: Write the Equation of the Ellipse
For a vertically oriented ellipse centered at (h, k) with semi-major axis a and semi-minor axis b, the equation is:
(frac{(x - h)^2}{b^2} frac{(y - k)^2}{a^2} 1)
Substituting h 0, k 0, a 17, and b 15 into the equation:
Final Equation:
(frac{x^2}{15^2} frac{y^2}{17^2} 1)
(frac{x^2}{225} frac{y^2}{289} 1)
The equation of the ellipse is thus (frac{x^2}{225} frac{y^2}{289} 1).
Conclusion
This step-by-step guide helps in determining the equation of an ellipse when the foci and the length of the major axis are known. Understanding these steps can be crucial in various mathematical and engineering applications.