Question

chapter 10 review
score: 75/100 answered: 8/10
question 10
given the ellipse $\frac{(x - 4)^2}{9}+\frac{(y - 2)^2}{36}=1$,
find the center point:
list the vertices (separated by a comma):
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Explanation:

Step1: Recall the standard form of an ellipse

The standard form of an ellipse is \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) (where \(a>b>0\)), and the center is \((h,k)\). For the given ellipse \(\frac{(x - 4)^2}{9}+\frac{(y - 2)^2}{36}=1\), we have \(h = 4\) and \(k=2\).

Step2: Find the vertices

Since \(a^2 = 36\), then \(a = 6\). The vertices of the ellipse \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) are \((h,k\pm a)\). Substituting \(h = 4\), \(k = 2\) and \(a=6\), we get \((4,2 + 6)\) and \((4,2-6)\).

Answer:

Center point: \((4,2)\)
Vertices: \((4,8),(4,-4)\)