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?

1 Answer
Jan 10, 2017

Given: (y - k)^2/a^2 + (x - h)^2/b^2 = 1" [1]"
Center: (h,k)
Foci:(h, k-sqrt(a^2 - b^2)) and (h, k+sqrt(a^2 - b^2))
major axis = 2a
minor axis = 2b

Explanation:

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