Find the Foci (x^2)/6-(y^2)/16=1
The question is asking for the determination of the coordinates of the foci for a given ellipse. The given equation is in the form of (x^2)/a^2 - (y^2)/b^2 = 1, which is the standard equation of a horizontal ellipse centered at the origin. The values of a^2 and b^2 are provided, and these values can be used to calculate the distance of the foci from the center of the ellipse along the major axis. The question requires the application of the formula for an ellipse's foci which is based on the relationship c^2 = a^2 + b^2 (for a hyperbola it would be c^2 = a^2 - b^2, but this equation represents an ellipse). The values of 'c' give the distance from the center to each focus, and since this is a horizontal ellipse, the foci will be found at (±c, 0) using the center as a reference point.
$\frac{x^{2}}{6} - \frac{y^{2}}{16} = 1$
Answer
Solution:
Step 1:
Rewrite the given equation to conform to the standard form where the equation equals $1$. The equation is already in the desired form: $\frac{x^{2}}{6} - \frac{y^{2}}{16} = 1$.
Step 2:
Identify the equation as representing a hyperbola. The general 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. For our equation, we have $h = 0$, $k = 0$, $a = \sqrt{6}$, and $b = 4$.
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 find $c$.
Step 4.2:
Plug in the values for $a$ and $b$: $c = \sqrt{(\sqrt{6})^{2} + 4^{2}}$.
Step 4.3:
Simplify the expression.
Step 4.3.1:
Square $\sqrt{6}$ to get $6$: $c = \sqrt{6 + 4^{2}}$.
Step 4.3.2:
Square $4$ to get $16$ and add to $6$: $c = \sqrt{6 + 16}$.
Step 4.3.3:
Combine the terms under the radical: $c = \sqrt{22}$.
Step 5:
Determine the coordinates of the foci.
Step 5.1:
The first focus is found by adding $c$ to $h$: $(h + c, k)$.
Step 5.2:
Insert the known values to get the first focus: $(\sqrt{22}, 0)$.
Step 5.3:
The second focus is found by subtracting $c$ from $h$: $(h - c, k)$.
Step 5.4:
Insert the known values to get the second focus: $(-\sqrt{22}, 0)$.
Step 5.5:
The foci of the hyperbola are given by $(h \pm c, k)$. There are two foci for a hyperbola: $(\sqrt{22}, 0)$ and $(-\sqrt{22}, 0)$.
Knowledge Notes:
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 a vertex on the transverse axis, and $b$ is the distance from the center to a vertex on the conjugate axis.
Foci of a Hyperbola: The foci of a hyperbola are located along the transverse axis, at a distance $c$ from the center, where $c = \sqrt{a^{2} + b^{2}}$. The coordinates of the foci are $(h \pm c, k)$ for a hyperbola with a horizontal transverse axis.
Simplifying Radical Expressions: When simplifying expressions involving radicals, remember that $\sqrt{a^{2}} = a$ for any non-negative real number $a$.
Exponent Rules: The power of a power rule states that $(a^{m})^{n} = a^{m \cdot n}$. This rule is used when simplifying expressions with exponents.
By following these steps and applying the relevant knowledge, we can find the foci of the given hyperbola.
