# How do you find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola #5x^2-4y^2-40x-16y-36=0#?

##### 1 Answer

Convert the equation into 1 of 2 standard forms:

Then the desired information can be obtained by observation.

#### Explanation:

Given:

Add

Move all of the x related terms (on the left) so that they are the first 3 terms:

Remove a factor of 5 from the first 3 terms and a factor of -4 from the remaining terms:

Set the x term in the right side of the pattern

Solve for h:

Substitute

Set the y term in the right side of the pattern

Substitute

Combine the terms on the right:

Divide both sides by 100:

Write the denominators as squares:

This matches equation [1]; the horizontal transverse axis type.

The center is:

From equation [5] we see that

The vertices are:

substitute in the values from equation [5]: gives us

respectively

The foci are:

Substitute in the values from equation [5] gives us:

The equations of the the asymptotes are:

Respectively: