Question 623675
  <pre><font face = "Tohoma" size = 3 color = "indigo"><b> 
Hi,  
x2 + 4y2 + 10x – 56y = -205   |completing the Square
 (x+5)^2 + 4(y-7)^2  = -205 + 25 + 196 = 16
{{{(x+5)^2/16 + (y-7)^2/4 =(x+5)^2/4^2 + (y-7)^2/2^2 = 1}}}   
C(-5,7) and V(-1,7),V(-9,7)  & V(-5,9), V(-5,5)
{{{drawing(300,300,   -10,10,-10,10,  arc(-5,7,8,4),
 grid(1),
circle(-5, 7,0.4),
circle(-1, 7,0.4),
graph( 300, 300, -10,10,-10,10))}}}
<u>See below descriptions of various conics                         </u>
Standard Form of an Equation of a Circle is {{{(x-h)^2 + (y-k)^2 = r^2}}} 
where Pt(h,k) is the center and r is the radius

 Standard Form of an Equation of an Ellipse is {{{(x-h)^2/a^2 + (y-k)^2/b^2 = 1 }}} 
where Pt(h,k) is the center. (a variable positioned to correspond with major axis)
 a and b  are the respective vertices distances from center
 and ±{{{sqrt(a^2-b^2)}}}are the foci distances from center: a > b

Standard Form of an Equation of an Hyperbola opening right and  left is:
  {{{(x-h)^2/a^2 - (y-k)^2/b^2 = 1}}} with C(h,k) and vertices 'a' units right and left of center  
Foci are {{{sqrt(a^2+b^2)}}} units right and left of center along y = k
& Asymptote Lines passing thru C(h,k), with slopes  m =  ± b/a 

Standard Form of an Equation of an Hyperbola opening up and down is:
  {{{(y-k)^2/b^2 - (x-h)^2/a^2 = 1}}} with C(h,k) and vertices 'b' units up and down from center 
Foci {{{sqrt(a^2+b^2)}}}units units up and down from center, along x = h
& Asymptote Lines passing thru C(h,k), with slopes m =  ± b/a

the vertex form of a Parabola opening up(a>0) or down(a<0), {{{y=a(x-h)^2 +k}}} where(h,k) is the vertex.
The standard form is {{{(x -h)^2 = 4p(y -k)}}}, where  the focus is (h,k + p)

the vertex form of a Parabola opening right(a>0) or left(a<0), {{{x=a(y-k)^2 +h}}} where(h,k) is the vertex.
The standard form is {{{(y -k)^2 = 4p(x -h)}}}, where  the focus is (h +p,k )