Question 838344
<pre><font face = "Tohoma" size = 3 color = "indigo"><b> 
Hi
BY DEFINITION (Need to know the Standard forms of the Conics: see below)
9x^2+3y^2=27  0r  y^2/9 + x^2/3 = 1  is an ellipse C(0,0): major axis along y-axis
vertices: (sqrt(3),0)& (sqrt(3),0) AND (0,3)&(0,-3) foci (0,sqrt(6)) and (0,-sqrt(6))  
| ±{{{sqrt(9-3)}}}are the foci distances from center: a > b
{{{drawing(300,300,   -6, 6, -6, 6, grid(1), arc(0,0,2sqrt(3),6), 
circle(0, 0,0.2),
circle(0, 3,0.2),
circle(0, -3,0.2),
circle(sqrt(3),0,0.2),
circle(-sqrt(3),0, 0.2),
circle(0, sqrt(6),0.2),
circle(0, -sqrt(6),0.2),
graph( 300, 300, -6, 6, -6, 6))}}}

<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 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,  2b the length of the transverse axis
Foci {{{sqrt(a^2+b^2)}}}units units up and down from center, along x = h
& Asymptotes Lines passing thru C(h,k), with slopes m =  ± b/a

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,   2a the length of the transverse axis
Foci are {{{sqrt(a^2+b^2)}}} units right and left of center along y = k
& Asymptotes 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  and  x = h  is the Line of Symmetry
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 and  y = k  is the Line of Symmetry
The standard form is {{{(y -k)^2 = 4p(x -h)}}}, where  the focus is (h +p,k )