Question

the graph shows an ellipse. write its equation in standard form.

Explanation:

Step1: Identify the center, major and minor axes

The center of the ellipse is at the origin \((0,0)\) since it's symmetric about both axes. From the graph, the ellipse extends 3 units along the x - axis (from \(x=- 3\) to \(x = 3\)) and 5 units along the y - axis (from \(y=-5\) to \(y = 5\)). So, \(a = 5\) (semi - major axis, along y - axis) and \(b=3\) (semi - minor axis, along x - axis). The standard form of an ellipse centered at the origin with a vertical major axis is \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) (when \(a>b\)).

Step2: Substitute the values of a and b

We know that \(a = 5\), so \(a^{2}=25\) and \(b = 3\), so \(b^{2}=9\). Substituting these values into the standard form equation \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\), we get \(\frac{x^{2}}{9}+\frac{y^{2}}{25}=1\).

Answer:

\(\frac{x^{2}}{9}+\frac{y^{2}}{25}=1\)