Question 334889
Find the equation of the ellipse with x intercepts at (±6, 0) and y intercepts at (0,±4) and find the foci.
<pre><font size = 4 color = "indigo"><b>

{{{drawing(400,2000/7, -7,7,-5,5,

graph(400,2000/7,-7,7,-5,5), blue(line(0,0,0,4)),

green(line(0,0,6,0)), locate(4.4,.7,F), locate(-4.5,.7,F),
red(line(2sqrt(5),-.2,2sqrt(5),.2),line(-2sqrt(5),-.2,-2sqrt(5),.2)),
arc(0,0,12,-8),
locate(6.2,.6,V), locate(-6.2,.6,V)



   )}}}

The vertices are marked with V's, the foci with F's

Ellipses which have this shape (like an egg sitting on a table,
not like the number "0") have this equation (learn it):

{{{x^2/a^2}}}{{{""+""}}}{{{y^2/b^2}}}{{{""=""}}}{{{1}}}


The green line is a, which is 6 units long, called the semi-major axis
The purple line is b, which is 4 units long, called the semi-minor axis,
so the equation of the ellipse is

{{{x^2/6^2}}}{{{""+""}}}{{{y^2/4^2}}}{{{""=""}}}{{{1}}}

or

{{{x^2/36}}}{{{""+""}}}{{{y^2/16}}}{{{""=""}}}{{{1}}}

The foci (±c,0) are found by this Pythagorean equation for all ellipses:

{{{c^2=a^2-b^2}}}
{{{c^2=6^2-4^2}}}
{{{c^2=36-15}}}
{{{c^2=20}}}
{{{c=sqrt(20)}}}
{{{c=sqrt(4*5)}}}
{{{c=2sqrt(5)}}}

The foci are {{{F1(2sqrt(5),0)}}} and {{{F2(-2sqrt(5),0)}}}

Edwin</pre></b.