Find equation of hyperbola with asymptotes slopes of +/-4 and foci of (4,0) and (-2,0)

Question

How do you find the equation of a hyperbola with the slopes of asymptotes of +/-4 and foci of (4,0) and (-2,0)?  I don't know how to find a and b but I was able to find c.

Answer 1

The asymptotes have a numerical slope of b/a=4 where a and b are given by:

x²/a²-y²/b²=1 and x and y are (0,0) centred.

The distance between the foci is 6 along the x-axis of the left-right hyperbola.

6=2√a²+b², so √a²+b²=3 and a²+b²=9=c² making the focus c=3 from the centre.

The mid point of the distance between the foci is the centre which is (1,0) because 4-1=1-(-2)=3=c.

Since b/a=4, b=4a and a²+b²=a²+16a²=9, a²=9/17 and b²=16*9/17=144/17.

The equation of the hyperbola is therefore 17(x-1)²/9-17y²/144=1.