Find the Foci (x^2)/9-y^2=1
The question asks you to determine the coordinates of the foci of a hyperbola defined by the equation (x^2)/9 - y^2 = 1. A hyperbola is a type of conic section that features two separate curves. Each curve has a focal point, and the foci (plural of focus) are key components in the definition and construction of the hyperbola. The equation provided is in the standard form of a hyperbola with the horizontal transverse axis, where the terms represent squares of the distances from the center to the vertices and co-vertices. To find the foci, one would typically use the relationship between the distances represented in the equation to calculate their exact positions relative to the center of the hyperbola.
$\frac{x^{2}}{9} - y^{2} = 1$
Answer
Solution:
Step 1:
Rewrite the given equation to match the standard form of a hyperbola equation, which requires the equation to be set equal to $1$.
$$\frac{x^{2}}{9} - \frac{y^{2}}{1} = 1$$
Step 2:
Identify the equation as a hyperbola. The standard form of a hyperbola is:
$$\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1$$
Step 3:
Determine the values of $h$, $k$, $a$, and $b$ by comparing the given equation to the standard form.
$$a = 3, b = 1, h = 0, k = 0$$
Step 4:
Calculate the distance $c$ from the center to a focus of the hyperbola.
Step 4.1:
Use the formula for the distance $c$:
$$c = \sqrt{a^{2} + b^{2}}$$
Step 4.2:
Plug in the values for $a$ and $b$:
$$c = \sqrt{3^{2} + 1^{2}}$$
Step 4.3:
Carry out the simplification process.
Step 4.3.1:
Square the number $3$:
$$c = \sqrt{9 + 1^{2}}$$
Step 4.3.2:
Recognize that any number raised to the power of $1$ remains the same:
$$c = \sqrt{9 + 1}$$
Step 4.3.3:
Add the numbers inside the square root:
$$c = \sqrt{10}$$
Step 5:
Locate the foci of the hyperbola.
Step 5.1:
The first focus is found by adding $c$ to $h$:
$$(h + c, k)$$
Step 5.2:
Substitute the known values and simplify:
$$(\sqrt{10}, 0)$$
Step 5.3:
The second focus is found by subtracting $c$ from $h$:
$$(h - c, k)$$
Step 5.4:
Substitute the known values and simplify:
$$(-\sqrt{10}, 0)$$
Step 5.5:
The foci of the hyperbola are given by the general form:
$$(h \pm c, k)$$ Thus, the foci are:
$$(\sqrt{10}, 0), (-\sqrt{10}, 0)$$
Knowledge Notes:
Hyperbolas are a type of conic section that can be represented by the equation:
$$\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1$$ for a hyperbola that opens left and right, or $$\frac{(y - k)^{2}}{a^{2}} - \frac{(x - h)^{2}}{b^{2}} = 1$$ for a hyperbola that opens up and down. Here, $(h, k)$ is the center of the hyperbola, $a$ is the distance from the center to the vertices along the transverse axis, and $b$ is the distance from the center to the vertices along the conjugate axis.
The distance $c$ from the center to each focus of the hyperbola is found using the relationship:
$$c = \sqrt{a^{2} + b^{2}}$$
The foci of a hyperbola are located along the transverse axis, which is the axis that passes through the vertices. The coordinates of the foci are given by:
$$(h + c, k) \text{ and } (h - c, k)$$ for a hyperbola that opens left and right, and $$(h, k + c) \text{ and } (h, k - c)$$ for a hyperbola that opens up and down.
In the context of the given problem, the hyperbola opens left and right because the $x^2$ term is positive and over $a^2$. The center of this hyperbola is at the origin (0,0), and the foci are located at a distance $c$ from the center along the x-axis.
