To write the equation of an ellipse in standard form, we need to understand the basic components of the ellipse equation and the given dimensions:
The standard form of an ellipse equation centered at the origin [tex]\((0, 0)\)[/tex] is:
[tex]\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\][/tex]
where:
- [tex]\(a\)[/tex] represents the semi-major axis
- [tex]\(b\)[/tex] represents the semi-minor axis
Given:
- The height of the ellipse is 50 units.
- The width of the ellipse is 40 units.
We need to determine the lengths of the semi-major axis (b) and the semi-minor axis (a).
1. Semi-major axis (b) Calculation:
- Height (major axis length) = 50 units
- Semi-major axis (b) = height / 2 = 50 / 2 = 25 units
2. Semi-minor axis (a) Calculation:
- Width (minor axis length) = 40 units
- Semi-minor axis (a) = width / 2 = 40 / 2 = 20 units
Using these values, we substitute into the standard ellipse equation:
3. Square the lengths of the axes:
- Semi-major axis (b): [tex]\(25^2 = 625\)[/tex]
- Semi-minor axis (a): [tex]\(20^2 = 400\)[/tex]
4. Form the standard ellipse equation:
[tex]\[
\frac{x^2}{20^2} + \frac{y^2}{25^2} = \frac{x^2}{400} + \frac{y^2}{625} = 1
\][/tex]
Therefore, the correct equation in standard form for the given ellipse is:
[tex]\[
\frac{x^2}{400} + \frac{y^2}{625} = 1
\][/tex]
So, the correct answer is:
[tex]\[
\frac{x^2}{400} + \frac{y^2}{625} = 1
\][/tex]
