# 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?

Dec 4, 2017

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:

${x}^{2} + 5 {y}^{2} + 4 x - 70 y + 209 = 0$

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

${x}^{2} + 4 x + 5 {y}^{2} - 70 y = - 209$

${\left(x - h\right)}^{2} = {x}^{2} - 2 h x + {h}^{2}$.

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:

${x}^{2} + 4 x + {h}^{2} + 5 {y}^{2} - 70 y = - 209 + {h}^{2}$

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:

$- 2 h x = 4 x$

$h = - 2$

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

${\left(x - \left(- 2\right)\right)}^{2} + 5 {y}^{2} - 70 y = - 209 + 4$

${\left(y - k\right)}^{2} = {y}^{2} - 2 k y + {k}^{2}$

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

$5 {\left(y - k\right)}^{2} = 5 {y}^{2} - 10 k y + 5 {k}^{2}$

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

${\left(x - \left(- 2\right)\right)}^{2} + 5 {y}^{2} - 70 y + 5 {k}^{2} = - 209 + 4 + 5 {k}^{2}$

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

$- 10 k y = - 70 y$

$k = 7$

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

${\left(x - \left(- 2\right)\right)}^{2} + 5 {\left(y - 7\right)}^{2} = - 209 + 4 + 245$

Simplify the right:

${\left(x - \left(- 2\right)\right)}^{2} + 5 {\left(y - 7\right)}^{2} = 40$

Divide both sides by 40:

${\left(x - \left(- 2\right)\right)}^{2} / 40 + {\left(y - 7\right)}^{2} / 8 = 1$

Write the denominators as squares:

${\left(x - \left(- 2\right)\right)}^{2} / {\left(2 \sqrt{10}\right)}^{2} + {\left(y - 7\right)}^{2} / {\left(2 \sqrt{2}\right)}^{2} = 1$

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:

$\left(h , k\right) = \left(- 2 , 5\right)$

The left focus is:

$\left(h - \sqrt{{a}^{2} - {b}^{2}} , k\right) = \left(- 2 - \sqrt{40 + 8} , 7\right)$

$\left(h - \sqrt{{a}^{2} - {b}^{2}} , k\right) = \left(- 2 - 4 \sqrt{3} , 7\right)$

The right focus is:

$\left(h + \sqrt{{a}^{2} - {b}^{2}} , k\right) = \left(- 2 + 4 \sqrt{3} , 7\right)$

The length of the major axis is:

$2 a = 4 \sqrt{10}$

The length of the minor axis is:

$2 b = 4 \sqrt{2}$