Question 38377
1) Find the vertices and foci of the ellipse
{{{4x^2 + 25y^2 - 16x + 100y = -16}}}
{{{4x^2-16x+25y^2+100y=-16}}}
{{{(4x^2-16x)+(25y^2+100y)=-16}}}
{{{4(x^2-4x)+25(y^2+4y)=-16}}}
{{{4(x-2)^2+25(y+2)^2=100}}}
{{{((x-2)^2/25)+((y+2)^2/4)=1}}}
The ellipse has a horizontal major axis. Center is (2,-2).
a=5
b=2
{{{c^2=a^2-b^2}}}
{{{c^2=5^2-2^2}}}
{{{c^2=25-4}}}
{{{c=(21)^(1/2)}}}
Vertices: (7,-2),(-3,-2),(2,0),(2,-4)
Foci: (2-(21)^(1/2),-2),(2+(21)^(1/2),-2)
2) Find the center and asymptotes of the hyperbola 
4x2 - 9y2 - 8x - 18y = 41
{{{4x^2-8x-9y^2-18y=41}}}
{{{(4x^2-8x)+(-9y^2-18y)=41}}}
{{{4(x^2-2x)-9(y^2+2y)=41}}}
{{{4(x-1)^2-9(y+1)^2=36}}}
{{{((x-1)^2/9)-((y+1)^2/4)=1}}}
This hyperbola has a horizontal transverse axis. The center is (1,-1). The slope of the asymptotes is b/a or -b/a. a=3 and b=2
{{{y+1=(2/3)(x-1)}}} or {{{y+1=(-2/3)(x-1)}}}
3) solve the system x^2 + y^2 = 16 and x^2/16 + y^2/9 = 1
for {{{x^2 + y^2 = 16}}}, {{{y^2 = -x^2 + 16}}}
Plug in: {{{x^2/16 + y^2/9 = 1}}}
{{{144(x^2/16) + 144((-x^2 + 16)/9) = 144(1)}}}
{{{9x^2 + 16(-x^2 + 16) = 144}}}
{{{9x^2 - 16x^2 = -112}}}
{{{-7x^2 = -112}}}
{{{x^2 = 16}}}
{{{x = (4) or (-4)}}}
Plug in:
{{{x^2 + y^2 = 16}}}
{{{(4)^2 + y^2 = 16}}}
{{{y^2 = 0}}}
{{{y = 0}}}
(4,0) and (-4,0) are your answers
4) find the center and radius of the circle 
{{{x^2 + y^2 -4x + 2y = 4}}}
{{{x^2 - 4x + y^2 + 2y = 4}}}
{{{(x^2 - 4x) + (y^2 + 2y) = 4}}}
{{{(x - 2)^2 + (y + 1)^2 = 9}}}
The center is (2,-1) and the radius is 3.