Write the equation of an ellipse with intercepts (-2,0), (2,0), (0,4), (0,-4). Then find the foci. Write the answer as square root (No decimals) Show work.
First, the major axis of the ellipse is vertical. The standard form of the equation of a vertical ellipse is [tex]\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1[/tex]. Here we can see that [tex]b=2[/tex] and [tex]a=4[/tex]. So the equation is [tex]\dfrac{x^2}{4}+\dfrac{y^2}{16}=1[/tex]. To find the foci, use [tex]c^2=a^2-b^2= 16-4\\ \implies c=\sqrt{12}=2\sqrt{3}[/tex]. The foci will be located on the axis the vertexes are on, so that makes the coordinates of the foci [tex](0,\pm 2\sqrt{3})[/tex].
