Question

what are the lengths of the major and minor axes of the ellipse ( 72y + 71 = -5x^{2}-6y^{2}-10x )?
write your answer in simplified, rationalized form.
length of major axis =
length of minor axis =

Explanation:

Step1: Rearrange the equation

Rearrange the given equation \(72y + 71=-5x^{2}-6y^{2}-10x\) to the standard form of an ellipse.
First, group the \(x\) - terms and \(y\) - terms:

$$5x^{2}+10x + 6y^{2}-72y=-71$$

Complete the square for the \(x\) - terms:

$$5(x^{2}+2x)=5(x^{2}+2x + 1-1)=5((x + 1)^{2}-1)=5(x + 1)^{2}-5$$

Complete the square for the \(y\) - terms:

$$6(y^{2}-12y)=6(y^{2}-12y+36 - 36)=6((y - 6)^{2}-36)=6(y - 6)^{2}-216$$

Substitute back into the equation:

$$5(x + 1)^{2}-5+6(y - 6)^{2}-216=-71$$

$$5(x + 1)^{2}+6(y - 6)^{2}=150$$

Divide by \(150\) to get the standard form \(\frac{(x + 1)^{2}}{30}+\frac{(y - 6)^{2}}{25}=1\)

Step2: Identify \(a\) and \(b\)

For the standard form of an ellipse \(\frac{(x - h)^{2}}{a^{2}}+\frac{(y - k)^{2}}{b^{2}} = 1\) (\(a>b\) for horizontal major axis or \(b > a\) for vertical major axis). Here \(a^{2}=30\), \(b^{2}=25\), so \(a=\sqrt{30}\), \(b = 5\)

Step3: Calculate the lengths of the axes

The length of the major axis is \(2a\) and the length of the minor axis is \(2b\)
If \(a=\sqrt{30}\), \(b = 5\), then the length of the major axis \(=2\sqrt{30}\) and the length of the minor axis \(=10\)

Answer:

Length of major axis \(=2\sqrt{30}\)
Length of minor axis \(=10\)