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):
question help: video 1 video 2 post to forum

Explanation:

Step1: Recall the standard form of an ellipse

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

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 \((h,k\pm a)\).
Substitute \(h = 4\), \(k = 2\) and \(a = 6\) into the formula.
When \(y=k + a\), \(y=2+6=8\), and the point is \((4,8)\).
When \(y=k - a\), \(y=2 - 6=-4\), and the point is \((4,-4)\).

Answer:

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