Question

PLEASE HELP!!! very confused

Answer 1

Answer:

Step-by-step explanation:

The vertices lie on the x-axis, as is determined by their coordinates. This makes the center of this hyperbola (0, 0) because the center is directly between the vertices. The fact that the foci also lie on the x-axis tells us that this is the main axis. What this also tells us is which way the hyperbola "opens". This one opens to the left and the right as opposed to up and down. The standard form for this hyperbola is:

$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$ and so far we have that h = 0 and k = 0.

By definition, a is the distance between the center and the vertices. So a = 5, and a-squared is 25. So we're getting there. Now here's the tricky part.

The expressions for the foci are (h-c, k) and (h+c, k). Since we know the foci lie at +/-13, we can use that to solve for c:

If h+c = 13 and h = 0, then

0 + c = 13 and c = 13.

We need that c value to help us find b:

$c^2=a^2+b^2$ and

$13^2=5^2+b^2$ and

$169=25+b^2$ and

$144=b^2$ so

b = 12. Now we're ready to fill in the equation:

$\frac{x^2}{25}-\frac{y^2}{144}=1$ and there you go!