How do you find the coordinates of the center, foci, the length of the major and minor axis given #3x^2+y^2+18x-2y+4=0#?

The following are the steps to put the given equation into the form of equation [1]:

Subtract 4 from both sides:

#3x^2 + y^2 + 18x - 2y = - 4" [2]"#

Group the x terms and the y terms together on the left:

#3x^2 + 18x + y^2 - 2y = - 4" [3]"#

Because the coefficient of the x^2 term is 3, add #3h^2 to both sides ; make it the 3rd term on the left and the first term on the right:

#3x^2 + 18x + 3h^2 + y^2 - 2y = 3h^2 - 4" [4]"#

Because the coefficient of the y^2 term is 1, add k^2 to both sides; make it the sixth term on the left and the second term on the right:

#3x^2 + 18x + 3h^2 + y^2 - 2y + k^2 = 3h^2 + k^2 - 4" [5]"#

Remove a common factor of 3 from the first 3 terms:

#3(x^2 + 6x + h^2) + y^2 - 2y + k^2 = 3h^2 + k^2 - 4" [6]"#

Use the pattern for #(x - h)^2 = x^2 - 2hx + h^2#.

Match the "-2hx" term in the pattern with the "6x" term in equation [6] and write the equation:

#-2hx = 6x#

Solve for h:

#h = -3#

This means that we can substitute #(x - -3)^2# for #(x^2 + 6x + h^2)# on the left side of equation [6] and -3 for h on the right:

#3(x - -3)^2 + y^2 - 2y + k^2 = 3(-3)^2 + k^2 - 4" [7]"#

Use the pattern for #(y - k)^2 = y^2 - 2ky + k^2#.

Match the "-2ky" term in the pattern with the "-2y" term in equation [7] and write the equation:

#-2ky = -2y#

Solve for k:

#k = 1#

This means that we can substitute #(y - 1)^2# for #y^2 -2y + k^2# on the left side of equation [7] and 1 for k on the right:

#3(x - -3)^2 + (y - 1)^2 = 3(-3)^2 + 1^2 - 4" [8]"#

Simplify the right:

#3(x - -3)^2 + (y - 1)^2 = 6" [9]"#

Divide both sides by 6:

#(x - -3)^2/2 + (y - 1)^2/6 = 1" [10]"#

Swap terms and write the denominators as squares:

#(y - 1)^2/(sqrt6)^2+(x - -3)^2/(sqrt2)^2 = 1" [11]"#

We have the form of equation [1]

#h = -3#
#k = 1#
#a = sqrt6#
#b = sqrt2#
#sqrt(a^2 - b^2) = sqrt(6 - 2) = sqrt(4) = 2#

Center: #(-3, 1)#
Foci: #(-3, 1-2) and (-3, 1 + 2) = (-3, -1) and (-3, 3)#
Major axis: #2sqrt6#
Minor axis: #2sqrt2#