The focal points of a hyperbola are (0,6) and (0,-6), and the point (5,6) is on one of its branches. Find coordinates for the points where the hyperbola intersects its major axis. Also find equations for the asymptotes, and use them to help you draw the curve.

the center is clearly at (0,0) so the equation is

y^2/a^2 - x^2/b^2 = 1
with c=6, so a^2+b^2 = 36
The given point says that
36/a^2 - 25/b^2 = 1
36/a^2 - 25/(36-a^2) = 1
a = 4 or 9
clearly a cannot be 9, so a=4 and b^2 = 20
Thus, the equation of the hyperbola is

y^2/16 - x^2/20 = 1
and the rest you can easily determine, right?

Well, well, well, it seems like we have a hyperbola on our hands! Let's dive right into it, shall we?

The center of the hyperbola is the midpoint between the two focal points, which in this case is (0,0). The distance between the center and each focal point is called the distance between the center and the focus, or c. In this case, c = 6.

The distance between the center and a point on the hyperbola branch is called the distance between the center and a point on the branch, or a. In this case, a = 5.

Now, the equation of a hyperbola with center (h,k), where a is the distance between the center and a point on the branch, b is the distance between the center and a point on the transverse axis, and c is the distance between the center and the focus, is given by:

(x - h)² / a² - (y - k)² / b² = 1

Plugging in the values we know, we get:

(x - 0)² / 5² - (y - 0)² / b² = 1

Simplifying that, we get:

x² / 25 - y² / b² = 1

Now, we know that the point (5,6) lies on the hyperbola branch, so let's plug that into the equation and solve for b:

(5)² / 25 - (6)² / b² = 1
25 / 25 - 36 / b² = 1
1 - 36 / b² = 1
-36 / b² = 0
b² = 36
b = 6

So, the equation of our hyperbola becomes:

x² / 25 - y² / 36 = 1

Now, as for the points where the hyperbola intersects its major axis, those would be the points (a, 0) and (-a, 0). In this case, a = 5, so those points would be (5, 0) and (-5, 0).

As for the asymptotes, they have equations of the form y = ± (b / a) * x. In this case, that would be y = ± (6 / 5) * x.

Now, with all this information, it's time to get drawin'! Plot the center at (0,0), the focal points at (0,6) and (0,-6), the points of intersection on the major axis at (5,0) and (-5,0), and finally, sketch in those asymptotes y = ± (6 / 5) * x.

Ta-da! You've got yourself a hyperbola, my friend. Enjoy your math artistry!

To find the coordinates where the hyperbola intersects its major axis, let's start by finding the equation of the hyperbola.

Since the focal points are (0, 6) and (0, -6), and the point (5, 6) lies on one of its branches, we can determine that the hyperbola is centered at the origin (0, 0) and has vertical transverse axis.

The general equation for a hyperbola with the center at the origin is:

(x^2 / a^2) - (y^2 / b^2) = 1,

where "a" is the distance from the center to a vertex, and "b" is the distance from the center to a co-vertex.

To find the value of "a", we can use the distance formula between the center and the given point on the branch:

a^2 = (x - h)^2 + (y - k)^2,
a^2 = (0 - 0)^2 + (6 - 0)^2,
a^2 = 0 + 36,
a^2 = 36,
a = √36,
a = 6.

So, we have the value of "a" as 6.

Now, let's find the value of "b" using the distance formula between the center and one of the foci:

c^2 = a^2 - b^2,
(6)^2 = (b)^2,
36 = b^2,
b = √36,
b = 6.

So, we have the value of "b" as 6.

Therefore, the equation for the hyperbola is:

(x^2 / 6^2) - (y^2 / 6^2) = 1,
(x^2 / 36) - (y^2 / 36) = 1.

To find the coordinates where the hyperbola intersects its major axis, we substitute y = 0 into the equation:

(x^2 / 36) - (0 / 36) = 1,
x^2 / 36 = 1,
x^2 = 36,
x = ± √36,
x = ± 6.

So, the hyperbola intersects its major axis at the points (6, 0) and (-6, 0).

To find the equations for the asymptotes, we use the formula:

y = ± (b / a) * x,
y = ± (6 / 6) * x,
y = ± x.

This gives us the equations for the asymptotes as y = x and y = -x.

To help draw the curve, we plot the center of the hyperbola at the origin (0, 0), and the two vertices at (6, 0) and (-6, 0). Then, we can draw the asymptotes y = x and y = -x. Finally, we can sketch the curves of the hyperbola using those reference points and the general shape of a hyperbola.

To find the coordinates where the hyperbola intersects its major axis, we need to determine the distance between the point (5,6) and the focal points (0,6) and (0,-6).

The distance between two points (x1, y1) and (x2, y2) is given by the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Let's calculate the distance between (5,6) and (0,6):

d1 = √((0 - 5)^2 + (6 - 6)^2)
= √((-5)^2 + 0)
= √25
= 5

Similarly, let's calculate the distance between (5,6) and (0,-6):

d2 = √((0 - 5)^2 + (-6 - 6)^2)
= √((-5)^2 + (-12)^2)
= √25 + 144
= √169
= 13

Since the distance from (5,6) to (0,6) is less than the distance from (5,6) to (0,-6), (5,6) lies on the branch of the hyperbola closer to the focal point (0,6).

Now, let's find the coordinates where the hyperbola intersects its major axis. Since the major axis passes through the focal points, the points of intersection will have the same y-coordinate as the focal points.

For the given hyperbola, the points of intersection on the major axis will be (x, ±6).

Next, let's find the equations for the asymptotes of the hyperbola. The asymptotes are straight lines that the hyperbola approaches but never touches. The equations of the asymptotes for a hyperbola with horizontal transverse axis (as in this case) are:

y = ±(a/b)(x - h) + k

where (h, k) is the center of the hyperbola and a and b are the lengths of the semi-major and semi-minor axes, respectively.

In our case, the center of the hyperbola is at the origin (0,0) since the focal points have the same x-coordinate.

The equation for the asymptote can be simplified to:

y = ±(a/b)x

Now, let's determine the values of a and b for the given hyperbola.

The distance between the center of the hyperbola (0,0) and one of the vertices on the major axis is called the semi-major axis (a). In this case, the distance is 6 units:

a = 6

The distance between the center of the hyperbola (0,0) and one of the points where the hyperbola intersects an asymptote is called the semi-minor axis (b). In this case, the distance is 5 units:

b = 5

Therefore, the equation for the asymptotes of the hyperbola is:

y = ±(6/5)x

Now that we have the coordinates for the points where the hyperbola intersects its major axis ((x, ±6)) and the equations for the asymptotes, we can draw the curve on a graph using this information.