Answer:
[tex]\dfrac{x^2}{81}+\dfrac{y^2}{16}=1[/tex]
Step-by-step explanation:
The given graph shows a horizontal ellipse.
A horizontal ellipse is one in which the major axis (the longest diameter) is parallel to the x-axis, and the minor axis (the shortest diameter) is parallel to the y-axis.
The general equation of a horizontal ellipse is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{General equation of a horizontal ellipse}}\\\\\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1\\\\\textsf{where:}\\\phantom{ww}\bullet \textsf{$(h,k)$ is the center.}\\\phantom{ww}\bullet\textsf{$2a$ is the major axis.}\\\phantom{ww}\bullet\textsf{$2b$ is the minor axis.}\\\phantom{ww}\bullet\textsf{$a$ is the major radius.}\\\phantom{ww}\bullet\textsf{$b$ is the minor radius.}\end{array}}[/tex]
From observation of the graph ellipse, the endpoints of the major axis are (-9, 0) and (9, 0), whilst the endpoints of the minor axis are (0,-4) and (0, 4).
The x-coordinate of the center (h) is the midpoint of the x-coordinates of the endpoints of the major axis, and the y-coordinate of the center (k) is the midpoint of the y-coordinates of the endpoints of the major axis. Therefore, h = 0 and k = 0, so the center of the ellipse is the origin (0, 0).
The major radius (a) is the distance between the x-coordinates of the center on one of the endpoints of the major axis:
[tex]a=|0-(-9)|=|9|=9[/tex]
The minor radius (b) is the distance between the y-coordinates of the center on one of the endpoints of the minor axis:
[tex]b=|0-(-4)|=|4|=4[/tex]
Substitute the values of a, b, h and k into the general equation of a horizontal ellipse:
[tex]\dfrac{(x-0)^2}{9^2}+\dfrac{(y-0)^2}{4^2}=1[/tex]
[tex]\dfrac{x^2}{81}+\dfrac{y^2}{16}=1[/tex]
Therefore, the equation of the graphed ellipse is:
[tex]\Large\boxed{\boxed{\dfrac{x^2}{81}+\dfrac{y^2}{16}=1}}[/tex]
[tex]\hrulefill[/tex]
Additional Notes
The equation written in the general form of a conic section is:
[tex]16x^2+81y^2-1296=0[/tex]
