Given:
[tex]\frac{(x-2)^2}{36}+\frac{(y+3)^2}{24}=1[/tex]Required:
Find the endpoints of the major axis and minor axis of the ellipse.
Explanation:
The standard equation of the ellipse is:
[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]Where h and k are the centers of the ellipse and a and b are the length of the axis.
Rewrite the given equation as:
[tex]\frac{(x-2)^2}{(6)^2}+\frac{(y+3)^2}{(2\sqrt{6})^2}=1[/tex]Compare the given equation with the standard equation we get
[tex]\begin{gathered} h=2,\text{ k=-3} \\ a=6,\text{ b=2}\sqrt{6} \end{gathered}[/tex]Since a>b so the coordinate x-axis will be a major axis and the y-axis will be a minor axis.
The coordinate of the major axis are:
[tex](h\pm a,k)=(2\pm6,-3)[/tex]Take + sign
[tex](2+6,-3)=(8,-3)[/tex]Take - sign
[tex](2-6,-3)=(-4,-3_)[/tex]The coordinates of the minor axis are:
[tex](h,k\pm b)=(2,-3\pm2\sqrt{6})[/tex]Take the + sign
[tex](2,-3+2\sqrt{6})[/tex]Take the - sign
[tex](2,-3-2\sqrt{6})[/tex]Final Answer:
The coordinates of the major axis are: (8,-3) and (-4,-3)
The coordinates of the minor axis are:
[tex](2,-3\pm2\sqrt{6})[/tex]