Question

what are the lengths of the major and minor axes of the ellipse \\( \frac { ( x + 3 ) ^ { 2 } } { 87 } + \frac { ( y + 9 ) ^ { 2 } } { 100 } = 1 \\)?
write your answer in simplified, rationalized form.
length of major axis =
length of minor axis =

Explanation:

Step1: Recall the standard form of an ellipse

The standard form of an ellipse is \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) (\(a>b>0\)), where the length of the major axis is \(2a\) and the length of the minor axis is \(2b\).
For the given ellipse \(\frac{(x + 3)^2}{87}+\frac{(y + 9)^2}{100}=1\), we have \(a^2 = 100\), \(b^2=87\).

Step2: Calculate the length of the major axis

Since \(a^2 = 100\), then \(a = 10\).
The length of the major axis is \(2a\).
\(2a=2\times10 = 20\).

Step3: Calculate the length of the minor axis

Since \(b^2 = 87\), then \(b=\sqrt{87}\).
The length of the minor axis is \(2b\).
\(2b = 2\sqrt{87}\).

Answer:

Length of major axis \(=20\)
Length of minor axis \(=2\sqrt{87}\)