Question 608603
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Hi
Note:
 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
vertices (-2,-4) and (8,-4),{{{highlight(Center(3,-4))}}} and  a focus at (6,-4)or 3 from center  and    y = -4, major axis
{{{(x-3)^2/5^2 + (y+4)^2/b^2 = 1 }}} and{{{sqrt(25-b^2)}}} = 3, b = 4
{{{(x-3)^2/5^2 + (y+4)^2/4^2 = 1 }}} OR {{{(x-3)^2/25 + (y+4)^2/16 = 1 }}}
{{{drawing(300,300,   -10,10,-10,10,  arc(3,-4,10,8),
 grid(1),
circle(3, -4,0.4),
circle(6, -4,0.4),
graph( 300, 300, -10,10,-10,10,0,-4))}}}


See below descriptions of various conics
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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 )