Find the Foci (y^2)/81-(x^2)/100=1
The question asks for the coordinates of the foci of a hyperbola that is described by the given equation (y^2)/81 - (x^2)/100 = 1. The foci are two points on the transverse axis of the hyperbola, which are located symmetrically with respect to the center of the hyperbola, at a certain distance from the center. This distance can be found using the equation of the hyperbola, based on the relationship between the lengths of its axes and the distance to the foci. To solve the problem, one would need to identify this hyperbola's semi-major and semi-minor axes, calculate the distance between the center and the foci (focal length), and then determine the coordinates of the foci using this information.
$\frac{y^{2}}{81} - \frac{x^{2}}{100} = 1$
Answer
Solution:
Step 1
Rewrite the given equation to conform to the standard form where the equation equals $1$.
$$\frac{y^{2}}{81} - \frac{x^{2}}{100} = 1$$
Step 2
Identify the equation as that of a hyperbola. The standard equation for a hyperbola is:
$$\frac{(y - k)^{2}}{a^{2}} - \frac{(x - h)^{2}}{b^{2}} = 1$$
Step 3
Align the given hyperbola's equation with the standard form to determine the constants $h$, $k$, $a$, and $b$.
$$a = 9, b = 10, k = 0, h = 0$$
Step 4
Calculate the distance $c$ from the center of the hyperbola to its foci.
Step 4.1
Use the formula for finding $c$ in a hyperbola:
$$c = \sqrt{a^{2} + b^{2}}$$
Step 4.2
Plug in the values for $a$ and $b$:
$$c = \sqrt{9^{2} + 10^{2}}$$
Step 4.3
Perform the calculations:
Step 4.3.1
Square $9$:
$$c = \sqrt{81 + 10^{2}}$$
Step 4.3.2
Square $10$:
$$c = \sqrt{81 + 100}$$
Step 4.3.3
Add the results:
$$c = \sqrt{181}$$
Step 5
Determine the coordinates of the foci.
Step 5.1
The first focus is found by adding $c$ to $k$:
$$(h, k + c)$$
Step 5.2
Insert the known values for $h$, $c$, and $k$ and simplify:
$$(0, \sqrt{181})$$
Step 5.3
The second focus is found by subtracting $c$ from $k$:
$$(h, k - c)$$
Step 5.4
Insert the known values for $h$, $c$, and $k$ and simplify:
$$(0, -\sqrt{181})$$
Step 5.5
The foci of a hyperbola are given by the formula $(h, k \pm c)$. A hyperbola has two foci:
$$(0, \sqrt{181}), (0, -\sqrt{181})$$
Step 6
The foci of the hyperbola are at $(0, \sqrt{181})$ and $(0, -\sqrt{181})$.
Knowledge Notes:
A hyperbola is a type of conic section that appears as two separate curves facing away from each other. Each curve is called a branch of the hyperbola.
The standard form of a hyperbola with a vertical transverse axis (opening up and down) is given by:
$$\frac{(y - k)^{2}}{a^{2}} - \frac{(x - h)^{2}}{b^{2}} = 1$$ where $(h, k)$ is the center of the hyperbola, $a$ is the distance from the center to the vertices along the y-axis, and $b$ is the distance from the center to the vertices along the x-axis.
The foci of a hyperbola are located along the transverse axis, and the distance from the center to each focus is denoted by $c$. The relationship between $a$, $b$, and $c$ is given by the equation $c = \sqrt{a^{2} + b^{2}}$.
The vertices of a hyperbola are the points where each branch is closest to the center, and they lie on the transverse axis at a distance $a$ from the center.
The asymptotes of a hyperbola are the lines that each branch of the hyperbola approaches but never reaches. They intersect at the center of the hyperbola and have slopes of $\pm \frac{a}{b}$ or $\pm \frac{b}{a}$ depending on the orientation of the hyperbola.
To find the foci of a hyperbola, one can use the formula $(h, k \pm c)$ for a hyperbola with a vertical transverse axis or $(h \pm c, k)$ for a hyperbola with a horizontal transverse axis.
