# 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#?

##### 1 Answer

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

Then the desired information can be obtained.

#### Explanation:

Given:

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

Please observe the pattern

To make the x terms look like the pattern, we must insert

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:

This allows us to substitute

Please observe the pattern

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

This means that we must add

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

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

Simplify the right:

Divide both sides by 40:

Write the denominators as squares:

Here is the corresponding form:

This allows us to obtain the desired information by observation.

The center is:

The left focus is:

The right focus is:

The length of the major axis is:

The length of the minor axis is: