Question 1183607
{{{4x^2+9y^2-16x+18y-11=0}}}
{{{4(x^2-4x)+9(y^2+2y)=11}}}
{{{4(x^2-4x+4)+9(y^2+2y+1)=11+4*4+9*1}}}
{{{4(x-2)^2+9(y+1)^2=11+16+9}}}
{{{4(x-2)^2+9(y+1)^2=36}}}
The equation above shows that that {{{system(x-2=0,y+1=0)}}} or {{{system(x=2,y=-1)}}} are axes (of symmetry) of the ellipse.
The center of the ellipse is the point where they intersect:
the point {{{P(2,-1)}}}
Your teacher may like to write the equation as {{{(x-2)^2/9+(y+1)^2/4=1}}} or {{{(x-2)^2/3^2+(y+1)^2/2^2=1}}} ,
dividing by {{{36}}} both sides of
{{{4(x-2)^2+9(y+1)^2=36}}} .
They are just equivalent equations for the same ellipse.
You could say they show more clearly that for all the points of the ellipse
{{{(x-2)^2/9<=1}}} --> {{{abs(x-2)<=3}}} and {{{(y+1)^2/4<=1}}} --> {{{abs(y+1)<=2}}}
They also show that the vertices of the ellipse are on the axes at {{{system(y=-1,x=2 +- 3)}}} and {{{system(x=2,y=-1 +- 2)}}} 
at {{{A(-1,-1)}}} , {{{B(5,-1)}}} , {{{C(2,1)}}} , and {{{D(2,-3)}}} .
The segment AB, on the "horizontal" {{{y=-1}}} axis, is called the major axis, because for this ellipse it is longer, going {{{a=3}}} units to left and right of the center of the ellipse.
The segment CD, on the "vertical" {{{x=2}}} axis, is called the minor axis, because for this ellipse it is shorter, going {{{b=2}}} units up and down from the center of the ellipse.
The foci are on the major axis at a distance {{{c}}} to both sides of the center, and we calculate {{{c}}} from {{{a^2=b^2+c^2}}} .
{{{c=sqrt(3^2-2^2)=sqrt(9-4)=sqrt(5)}}} .
Soo the coordinates of the foci are {{{system(y=-1, x=2 +-sqrt(5))}}} .
The ends of the latera recta are the points on the ellipse with the same {{{x}}}{{{"="}}}{{{2 +-sqrt(5)}}}{{{"="}}}{{{approximately}}}{{{2+- 2.236}}} as the foci.
For those points {{{x-2=" " +-sqrt(5)}}} , so {{{(x-2)^2=5}}}
The {{{y}}} coordinates for those points can be calculated from {{{4(x-2)^2+9(y+1)^2=36}}} as
{{{4*5+9(y+1)^2=36}}}-->{{{20+9(y+1)^2=36}}}-->{{{9(y+1)^2=16}}}-->{{{(y+1)^2=16/9}}}-->{{{y+1=" " +-sqrt(16/9)}}}-->{{{y=-1 +- 4/3}}}
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