Finding the vertices of an ellipse is straightforward once you understand its standard form. The equation of an ellipse can be written in two ways depending on its orientation:
- For a horizontally oriented ellipse: (x – h)²/a² + (y – k)²/b² = 1
- For a vertically oriented ellipse: (x – h)²/b² + (y – k)²/a² = 1
In these equations:
- $(h, k)$ is the center of the ellipse,
- a is the distance from the center to the vertices along the major axis, and
- b is the distance from the center to the vertices along the minor axis.
To find the vertices:
- Identify the Center: From the equation, determine the values of h and k to find the center of the ellipse.
- Determine ‘a’ and ‘b’: Look for the values of a and b in the equation. Remember that a is associated with the major axis and b with the minor axis.
- Calculate the Vertices:
- If it is horizontally oriented ((x – h)²/a² + (y – k)²/b² = 1):
- Vertices are at: (h ± a, k)
- If it is vertically oriented ((x – h)²/b² + (y – k)²/a² = 1):
- Vertices are at: (h, k ± a)
By following these steps, you can easily identify the vertices of an ellipse and gain a better understanding of its shape and orientation.