Question 610808
  <pre><font face = "Tohoma" size = 3 color = "indigo"><b> 
Hi, 
 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 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
 5x^2 + 2y^2 = 10
{{{(x)^2/2 + (y)^2/5 = 1 }}}   ||C(0,0), a = {{{sqrt(2)}}} and b={{{sqr(5)}}}   
( 0, ±{{{sqrt(5)}}}) and ±({{{sqrt(2)}}},0)are the Vertices
Foci: {{{sqrt(5-2) = sqrt(3)}}} ( 0, ±{{{sqrt(3)}}}) 
{{{drawing(300,300,   -6, 6, -6, 6,   arc(0,0,2.81,4.47),
 grid(1),
circle(1.414, 0,0.2),
circle(0, 2.236,0.2),
circle(0, 1.732,0.2),
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 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}}} where Pt(h,k) is a center  with vertices 'a' units right and left of center.

Standard Form of an Equation of an Hyperbola opening up and down is:
  {{{(y-k)^2/b^2 - (x-h)^2/a^2 = 1}}} where Pt(h,k) is a center  with vertices 'b' units up and down from center.

the vertex form of a parabola opening up or down, {{{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 or left, {{{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 )