Question 1190960
 Write the equation of the ELLIPSE that satisfies the given conditions

given:

Vertices ({{{-5}}},{{{0}}}) and ({{{5}}},{{{0}}}) ->shows common {{{y}}} coordinate {{{0}}}. 

So {{{x }}}axis is major axis and {{{y}}} axis is minor axis, therefore{{{ a}}} greater than {{{b}}}
 

Standard equation for this ellipse is:


{{{(x - h)^2/a^2 + (y - k)^2 /b^2= 1}}}


the center is half way between vertices: {{{C }}}({{{(-5+5)/2}}},{{{(0+0)/2}}})=({{{0}}}.{{{0}}})=>{{{h=0}}} and {{{k=0}}}

{{{2a }}} is the distance between vertices and center=>{{{2a=10}}}->{{{a=5}}}

length of latus rectum {{{8/5}}}-> In this ellipse latus rectum is {{{8/5}}} means {{{2b^2/a=8/5}}}

so, {{{2b^2=8}}} ->{{{b^2=4 }}}

and {{{a=5}}} units


your equation is:


{{{(x - 0)^2/5^2 + (y - 0)^2 /4= 1}}}


{{{x^2/25 + y^2 /4= 1}}}



{{{ drawing(600, 600, -10, 10, -10, 10,
circle(-5,0,.12), locate(-5.3,0.5,V(-5,0)),
circle(5,0,.12), locate(5.2,0.5,V(5,0)),
graph(600, 600, -10, 10, -10, 10,  -(2/5)sqrt(25-x^2),(2sqrt(25-x^2))/5)) }}}