The equation of the ellipse with x-intercepts at (4,0) and (-4,0), y-intercepts at (0,9) and (0,-9) and center at (0,0) is [tex]\dfrac{x^2}{16} + \dfrac{y^2}{81^2}=1[/tex].
The equation of the ellipse is written as,
[tex]\dfrac{(x-h)^2}{b^2} + \dfrac{(y-k)^2}{a^2} = 1[/tex]
Where b and a are the minor and major radius of the ellipse, and (h,k) is the coordinate of the center of the ellipse.
As it is given that the center of the ellipse is at the center or at the origin. therefore, the coordinates of the center of the ellipse are at the point (0, 0).
Also, it is given that the x-intercepts are at (4,0) and (-4,0) while y-intercepts are at (0,9) and (0,-9). therefore, the minor radius of the ellipse is 4 while the major radius of the ellipse is 9.
Now, we know the general equation for the ellipse, therefore, the equation can be written as,
[tex]\dfrac{(x-h)^2}{b^2} + \dfrac{(y-k)^2}{a^2} = 1\\\\\dfrac{(x-0)^2}{4^2} + \dfrac{(y-0)^2}{9^2}=1\\\\\dfrac{x^2}{16} + \dfrac{y^2}{81^2}=1[/tex]
Hence, the equation of the ellipse with x-intercepts at (4,0) and (-4,0), y-intercepts at (0,9) and (0,-9) and center at (0,0) is [tex]\dfrac{x^2}{16} + \dfrac{y^2}{81^2}=1[/tex].
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