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

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.

1. 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.

2. 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 $c = \sqrt{a^{2} + b^{2}}$.

3. 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.

4. 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.