Answer:
[tex]e=0.66[/tex]
Step-by-step explanation:
The eccentricity of an ellipse is given by: [tex]e=\sqrt{1-\frac{b^2}{a^2} }[/tex]
The given ellipse has equation: [tex]9x^2+16y^2-72x+64y-368=0[/tex]
We can rewrite this equation in standard form to obtain:
[tex]\frac{(x-4)^2}{8^2}+\frac{y+2}{6^2}=1[/tex]
We compare to the general standard form equation: [tex]\frac{(x-h)^2}{a^2}+\frac{y-k}{b^2}=1[/tex]
to get: [tex]a=8,b=6[/tex]
We substitute into the eccentricity formula to get:
[tex]e=\sqrt{1-\frac{6^2}{8^2} }[/tex]
[tex]e=\frac{\sqrt{7}}{4}=0.661438[/tex]
The eccentricity is 0.66 to the nearest hundredth
