Question 391784
From the equation we can see<ul><li>that the center of the hyberbola is (6, -6).</li><li>that the major axis of the hyperbola is horizontal because the {{{x^2}}} term is positive (and the {{{y^2}}} term is negative (or subtracted)).</li><li>that {{{a^2 = 16}}} and {{{b^2 = 4}}}</li></ul>
"c" is the distance from the center to the foci along the major axis. So to find the foci we will need to find "c". The equation for hyperbolas that connects a, b and c is:
{{{c^2 = a^2 + b^2}}}
Using the values we already have for {{{a^2}}} and {{{b^2}}} this equation becomes:
{{{c^2 = 16 + 4}}}
which simplifies to
{{{c^2 = 20}}}
Finding the square root of each side we get:
{{{c = sqrt(20)}}}
which simplifies as follows:
{{{c = sqrt(4*5)}}}
{{{c = sqrt(4)*sqrt(5)}}}
{{{c = 2*sqrt(5)}}}<br>
So the foci are a distance of {{{2sqrt(5)}}}, horizontally and in both directions, from the center. Since we want to move hoziontally from the center, in both directions, we will both add and subtract {{{2sqrt(5)}}} to/from the x-coordinate of the center. So the foci will be:
({{{6 + 2sqrt(5)}}}, -6) and ({{{6 - 2sqrt(5)}}}, -6)