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#?
Convert the equation into 1 of 2 standard forms:
Then the desired information can be obtained by observation.
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:
Set the y term in the right side of the pattern
Combine the terms on the right:
Divide both sides by 100:
Write the denominators as squares:
This matches equation ; the horizontal transverse axis type.
The center is:
From equation  we see that
The vertices are:
substitute in the values from equation : gives us
The foci are:
Substitute in the values from equation  gives us:
The equations of the the asymptotes are: