Find the Foci (x^2)/9-(y^2)/16=1

The provided equation is the standard form of a hyperbola centered at the origin, where the terms involving x^2 and y^2 have different signs, indicating the hyperbola opens along the x-axis. The question is asking for the coordinates of the foci, which are specific points related to the hyperbola's shape and orientation. These points lie along the major axis of the hyperbola and are at an equal distance from the center. To find the foci, one typically uses the hyperbola's defining properties and its relationship to the lengths of its axes.

$\frac{x^{2}}{9} - \frac{y^{2}}{16} = 1$

Answer

Expert–verified

Solution:

Step 1:

Rewrite the equation to conform to the standard form where the right side equals $1$ . The equation should look like this: $\frac{x^{2}}{9} - \frac{y^{2}}{16} = 1$ .

Step 2:

Recognize that the equation represents a hyperbola. The general form for a hyperbola centered at $(h, k)$ is $\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1$ .

Step 3:

Identify the values of $h$ , $k$ , $a$ , and $b$ by comparing the given equation to the standard form. For our equation, we have:

$a = 3$
$b = 4$
$h = 0$
$k = 0$

Step 4:

Calculate the distance $c$ from the center to a focus of the hyperbola.

Step 4.1:

Use the formula $c = \sqrt{a^{2} + b^{2}}$ to determine $c$ .

Step 4.2:

Plug in the known values for $a$ and $b$ : $c = \sqrt{3^{2} + 4^{2}}$ .

Step 4.3:

Perform the calculations:

Step 4.3.1:

Square the value of $3$ : $c = \sqrt{9 + 4^{2}}$ .

Step 4.3.2:

Square the value of $4$ : $c = \sqrt{9 + 16}$ .

Step 4.3.3:

Add the results: $c = \sqrt{25}$ .

Step 4.3.4:

Express $25$ as a square of $5$ : $c = \sqrt{5^{2}}$ .

Step 4.3.5:

Extract the square root: $c = 5$ .

Step 5:

Determine the coordinates of the foci.

Step 5.1:

To find the first focus, add $c$ to $h$ : $(h + c, k)$ .

Step 5.2:

Insert the values for $h$ , $c$ , and $k$ : $(0 + 5, 0)$ .

Step 5.3:

To find the second focus, subtract $c$ from $h$ : $(h - c, k)$ .

Step 5.4:

Insert the values for $h$ , $c$ , and $k$ : $(0 - 5, 0)$ .

Step 5.5:

The foci of the hyperbola are given by $(h \pm c, k)$ . There are two foci for a hyperbola: $(5, 0)$ and $(- 5, 0)$ .

Step 6:

The foci of the hyperbola are $(5, 0)$ and $(- 5, 0)$ .

Knowledge Notes:

The problem involves finding the foci of a hyperbola, which is a type of conic section. A hyperbola is defined as the set of all points in a plane where the absolute difference of the distances from two fixed points (the foci) is constant.

Standard Form of a Hyperbola: The standard form of a hyperbola with a horizontal transverse axis is $\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1$ , where $(h, k)$ is the center of the hyperbola, $a$ is the distance from the center to the vertices along the x-axis, and $b$ is the distance from the center to the vertices along the y-axis.
Foci of a Hyperbola: The foci of a hyperbola are located along the transverse axis, which is the axis that passes through the vertices. The distance from the center to each focus is denoted by $c$ , and it is calculated using the formula $c = \sqrt{a^{2} + b^{2}}$ .
Vertices of a Hyperbola: The vertices are the points where the hyperbola intersects its transverse axis. For the given hyperbola, the vertices are at $(\pm a, 0)$ when centered at the origin.
Asymptotes of a Hyperbola: Asymptotes are lines that the hyperbola approaches but never intersects. The equations of the asymptotes for a hyperbola centered at the origin with a horizontal transverse axis are $y = \pm \frac{b}{a} x$ .

Understanding these concepts is crucial for solving problems related to hyperbolas, including finding the foci, vertices, asymptotes, and graphing the hyperbola.