Find the Foci (x^2)/16-(y^2)/64=1
The problem is asking to determine the coordinates of the foci of a hyperbola. A hyperbola is a type of conic section that can be defined as the set of all points in the plane where the absolute difference of the distances from any point on the hyperbola to two fixed points (the foci) is constant.
In this specific equation $(x^2)/16 - (y^2)/64 = 1$, it takes the standard form of a hyperbola equation $(x^2)/a^2 - (y^2)/b^2 = 1$, where $a$and $b$are real numbers that determine the shape and orientation of the hyperbola.
The question requires identifying the values of $a$and $b$from the given equation and then using them to calculate the positions of the foci, which are located along the major axis (horizontal axis, in this case) at a certain distance from the center of the hyperbola. This distance from the center to each focus is found using the relationship $c^2 = a^2 + b^2$, where $c$is the distance from the center to a focus. The foci would have coordinates of the form $(±c, 0)$, considering that the center of the hyperbola is at the origin (0,0).
$\frac{x^{2}}{16} - \frac{y^{2}}{64} = 1$
Answer
Solution:
Step 1:
Rewrite the equation to have 1 on the right side, adhering to the standard form of an ellipse or hyperbola equation.
$$\frac{x^{2}}{16} - \frac{y^{2}}{64} = 1$$
Step 2:
Recognize the equation as a hyperbola based on its form.
$$\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1$$
Step 3:
Identify the values of $h$, $k$, $a$, and $b$ by comparing the given equation to the standard form.
$$a = 4, b = 8, h = 0, k = 0$$
Step 4:
Calculate the distance $c$ from the center to a focus of the hyperbola.
Step 4.1:
Use the formula to find $c$.
$$c = \sqrt{a^{2} + b^{2}}$$
Step 4.2:
Insert the known values for $a$ and $b$ into the formula.
$$c = \sqrt{4^{2} + 8^{2}}$$
Step 4.3:
Perform the simplification process.
Step 4.3.1:
Square the number $4$.
$$c = \sqrt{16 + 8^{2}}$$
Step 4.3.2:
Square the number $8$.
$$c = \sqrt{16 + 64}$$
Step 4.3.3:
Combine the numbers under the radical.
$$c = \sqrt{80}$$
Step 4.3.4:
Express 80 as a product of squares to simplify the radical.
Step 4.3.4.1:
Factor out 16 from 80.
$$c = \sqrt{16 \cdot 5}$$
Step 4.3.4.2:
Recognize that $16$ is $4^{2}$.
$$c = \sqrt{4^{2} \cdot 5}$$
Step 4.3.5:
Simplify the radical by taking out the square term.
$$c = 4\sqrt{5}$$
Step 5:
Determine the coordinates of the foci.
Step 5.1:
Calculate the first focus by adding $c$ to $h$.
$$(h + c, k)$$
Step 5.2:
Insert the values for $h$, $c$, and $k$ and simplify.
$$(0 + 4\sqrt{5}, 0)$$
Step 5.3:
Calculate the second focus by subtracting $c$ from $h$.
$$(h - c, k)$$
Step 5.4:
Insert the values for $h$, $c$, and $k$ and simplify.
$$(0 - 4\sqrt{5}, 0)$$
Step 5.5:
Conclude that the foci of the hyperbola are given by the coordinates:
$$(4\sqrt{5}, 0), (-4\sqrt{5}, 0)$$
Step 6:
The foci of the hyperbola are located at the points $(4\sqrt{5}, 0)$ and $(-4\sqrt{5}, 0)$.
Knowledge Notes:
The problem involves finding the foci of a hyperbola, which is a type of conic section. The standard form of a hyperbola with a horizontal transverse axis is given by:
$$\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1$$ where $(h, k)$ is the center of the hyperbola, $a$ is the distance from the center to the vertices on the transverse axis, and $b$ is the distance from the center to the vertices on the conjugate axis. The distance $c$ from the center to each focus is found using the relationship $c = \sqrt{a^{2} + b^{2}}$. The foci of the hyperbola are located at $(h \pm c, k)$. In this problem, the given hyperbola is already in standard form, and we use the coefficients of $x^{2}$ and $y^{2}$ to find $a$ and $b$, respectively. After finding $c$, we use it to determine the coordinates of the foci.
