SOLUTION: i need to find the standard form of this equation and the properties of the hyperbola , i would really appreciate this help {{{ 12y^2-8x^2-120y+48x-204=0 }}}

 

Hi,

  ||Acquiring Standard form by completing the Square

 

 

  C = (3,5) Opening Up and down along x= 3

  V(3, 11) and V(3,-1) and F(3, 5 ± 3sqrt(10))

See below descriptions of various conics                         

Standard Form of an Equation of a Circle is  

where Pt(h,k) is the center and r is the radius



 Standard Form of an Equation of an Ellipse is  

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 ±are the foci distances from center: a > b



Standard Form of an Equation of an Hyperbola opening right and  left is:

   where Pt(h,k) is a center  with vertices 'a' units right and left of center 

and foci  units right and left of center along y = k



Standard Form of an Equation of an Hyperbola opening up and down is:

   where Pt(h,k) is a center  with vertices 'b' units up and down from center, 

and foci units units up and down from center, along x = h



the vertex form of a Parabola opening up(a>0) or down(a<0),  where(h,k) is the vertex.

The standard form is , where  the focus is (h,k + p)



the vertex form of a Parabola opening right(a>0) or left(a<0),  where(h,k) is the vertex.

The standard form is , where  the focus is (h +p,k )