Question

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`?

Answer 1

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`