Question

How do you find the coordinates of the center, foci, the length of the major and minor axis given `x^2+5y^2+4x-70y+209=0`?

Answer 1

Complete the squares so that the equation fits one these two forms:

`(x-h)^2/a^2+(y-k)^2/b^2= 1; a > b" [1]"`
`(x-h)^2/b^2+(y-k)^2/a^2= 1; a > b" [2]"`

Then the desired information can be obtained.

Explanation:

Given:

`x^2+5y^2+4x-70y+209=0`

Group, the x terms together, the y terms together, and move the constant term to the right:

`x^2+4x +5y^2-70y= -209`

Please observe the pattern

`(x-h)^2 = x^2-2hx + h^2`.

To make the x terms look like the pattern, we must insert `+h^2` on the left but, to maintain equality, we must, also add `h^2` to the right:

`x^2+4x+ h^2 +5y^2-70y= -209+ h^2`

We can solve for the value of h, if we set the middle term in the pattern equal to the middle term in the equation:

`-2hx = 4x`

`h = -2`

This allows us to substitute `(x - (-2))^2` for the x terms on the left and 4 for `h^2` on the right:

`(x-(-2))^2 +5y^2-70y= -209+ 4`

Please observe the pattern

`(y-k)^2 = y^2-2ky+k^2`

We must multiply both sides by 5 so that the pattern matches the equation:

`5(y-k)^2 = 5y^2-10ky+5k^2`

This means that we must add `5k^2` to both sides of the equation:

`(x-(-2))^2 +5y^2-70y+5k^2= -209+ 4+ 5k^2`

As we did with h, we can use the middle terms to solve for k:

`-10ky = -70y`

`k = 7`

This means that we can substitute 5(y-7)^2 for 5y^2-70y+5k^2 and 245 for `5k^2` on the right:

`(x-(-2))^2 +5(y-7)^2= -209+ 4+ 245`

Simplify the right:

`(x-(-2))^2 +5(y-7)^2= 40`

Divide both sides by 40:

`(x-(-2))^2/40 +(y-7)^2/8= 1`

Write the denominators as squares:

`(x-(-2))^2/(2sqrt10)^2 +(y-7)^2/(2sqrt2)^2= 1`

Here is the corresponding form:

`(x-h)^2/a^2+(y-k)^2/b^2= 1; a > b" [1]"`

This allows us to obtain the desired information by observation.

The center is:

`(h,k) = (-2,5)`

The left focus is:

`(h - sqrt(a^2-b^2), k) = (-2-sqrt(40+8), 7)`

`(h - sqrt(a^2-b^2), k) = (-2-4sqrt3, 7)`

The right focus is:

`(h + sqrt(a^2-b^2), k) = (-2+4sqrt3, 7)`

The length of the major axis is:

`2a = 4sqrt10`

The length of the minor axis is:

`2b = 4sqrt2`