An ellipse centered at the origin (0,0) can be represented by the standard form of its equation. The general equation of an ellipse in standard form is:
1. For a horizontally oriented ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
2. For a vertically oriented ellipse: \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \)
In this case, the vertex is located at (7, 0), meaning the semi-major axis is along the x-axis. Here, the distance from the center to the vertex defines the value of \(a\):
- \( a = 7 \)
Additionally, the co-vertex is located at (0, 5), which indicates the length of the semi-minor axis:
- \( b = 5 \)
Now, substituting the values of \(a\) and \(b\) into the standard form for a horizontally oriented ellipse, we get:
\( \frac{x^2}{7^2} + \frac{y^2}{5^2} = 1 \)
Which simplifies to:
\( \frac{x^2}{49} + \frac{y^2}{25} = 1 \)
This is the equation of the ellipse in standard form centered at the origin, with the given vertices. Remember, understanding the properties of ellipses can help visualize their geometric significance and can be really helpful in applications such as physics, astronomy, and engineering!