Question

assignment 10.1: the ellipse
score: 0/100 answered: 0/5
question 1
give the equation for the ellipse graphed above.
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Explanation:

Step1: Determine the standard form of the ellipse equation

The standard form of an ellipse centered at the origin \((h = 0,k = 0)\) is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\) (for a horizontal major axis) or \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) (for a vertical major axis). The length of the major axis is \(2a\) and the length of the minor axis is \(2b\).

Step2: Find the values of \(a\) and \(b\)

From the graph, the ellipse extends from \(x=-9\) to \(x = 9\), so \(a=9\) (semi - major axis). It extends from \(y=-5\) to \(y = 5\), so \(b = 5\) (semi - minor axis). Since \(a>b\), the major axis is horizontal.

Step3: Substitute \(a\) and \(b\) into the standard form

Substituting \(a = 9\) (\(a^{2}=81\)) and \(b = 5\) (\(b^{2}=25\)) into the standard form \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), we get \(\frac{x^{2}}{81}+\frac{y^{2}}{25}=1\)

Answer:

\(\frac{x^{2}}{81}+\frac{y^{2}}{25}=1\)