Question 608650
  <pre><font face = "Tohoma Ital" size = 3 color = "indigo"><b> 
Hi
-x^2 + y^2 - 6x -10y +17=0
x^2 - y^2 + 6x +10y -17=0  ||multiplying thru by -1 
(x+3)^2 -9 -(y-5)^2+25-17=0
{{{(x+3)^2 -(y-5)^2=1 }}} |Note:  C(-3,5) a=1 and b = 1
Note: 
 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.
a.find the vertices: with Center(-3,5)and a=1..V(-4,5) and V(-2,5)
b.find the length of the focal radius {{{sqrt(a^2+b^2)= highlight(sqrt(2))}}}
c.find the slopes of the asymptotes  m = ±b/a = ±  {{{ 1}}}
d.find the length of the conjugate axis: 2b is the conjugate axis length = {{{2}}} 

{{{drawing(300,300,-10,10,-10,10,  grid(1),
circle(-4, 5,0.3),
circle(-2, 5,0.3),
circle(-6.4, 0,0.3),
circle(6.4, 0,0.3),
graph(300,300,-10,10,-10,10,0,x+8,-x+2, sqrt((x+3)^2 -1)+5,-sqrt((x+3)^2-1)+5))}}}
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 )