Find the Foci (y^2)/9-(x^2)/36=1
The given problem is asking to determine the foci of an ellipse represented by the equation (y^2)/9 - (x^2)/36 = 1. An ellipse is defined as the set of all points in the plane where the sum of the distances to two fixed points, called foci, is constant. The equation given is in the standard form for the equation of an ellipse that is oriented along the Cartesian axes. The problem requires the application of knowledge regarding the properties of an ellipse, specifically how to identify the lengths of the major and minor axes from the equation, and then to use these lengths to calculate the coordinates of the foci.
$\frac{y^{2}}{9} - \frac{x^{2}}{36} = 1$
Answer
Solution:
Step 1
Rewrite the given equation to match the standard form where the right side equals $1$. The equation is already in the required form: $\frac{y^{2}}{9} - \frac{x^{2}}{36} = 1$.
Step 2
Recognize that the equation represents a hyperbola. The general form for a hyperbola centered at $(h, k)$ is $\frac{(y - k)^{2}}{a^{2}} - \frac{(x - h)^{2}}{b^{2}} = 1$.
Step 3
Identify the values of $h$, $k$, $a$, and $b$ by comparing the given equation to the standard form. For our hyperbola, we have:
- $a = 3$
- $b = 6$
- $k = 0$
- $h = 0$
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
Insert the known values for $a$ and $b$ into the formula: $c = \sqrt{3^{2} + 6^{2}}$.
Step 4.3
Perform the calculations:
Step 4.3.1
Square $3$: $c = \sqrt{9 + 6^{2}}$.
Step 4.3.2
Square $6$: $c = \sqrt{9 + 36}$.
Step 4.3.3
Add the results: $c = \sqrt{45}$.
Step 4.3.4
Express $45$ as a product of its prime factors: $c = \sqrt{3^{2} \cdot 5}$.
Step 4.3.5
Simplify the square root: $c = 3\sqrt{5}$.
Step 5
Determine the coordinates of the foci.
Step 5.1
The coordinate for the first focus is $(h, k + c)$.
Step 5.2
Substitute the values for $h$, $k$, and $c$: First focus is at $(0, 3\sqrt{5})$.
Step 5.3
The coordinate for the second focus is $(h, k - c)$.
Step 5.4
Substitute the values for $h$, $k$, and $c$: Second focus is at $(0, -3\sqrt{5})$.
Step 5.5
The foci of the hyperbola are at $(0, 3\sqrt{5})$ and $(0, -3\sqrt{5})$.
Step 6
The foci of the hyperbola are $(0, 3\sqrt{5})$ and $(0, -3\sqrt{5})$.
Knowledge Notes:
The problem involves finding the foci of a hyperbola. Relevant knowledge points include:
Standard Form of a Hyperbola: The equation of a hyperbola in standard form is $\frac{(y - k)^{2}}{a^{2}} - \frac{(x - h)^{2}}{b^{2}} = 1$ for a vertical hyperbola or $\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1$ for a horizontal hyperbola, where $(h, k)$ is the center of the hyperbola.
Components of a Hyperbola: In the standard form, $a$ is the distance from the center to the vertices along the axis of symmetry, $b$ is the distance from the center to the co-vertices, and $c$ is the distance from the center to the foci.
Relationship between $a$, $b$, and $c$: For hyperbolas, $c^2 = a^2 + b^2$.
Foci of a Hyperbola: The foci are located along the axis of symmetry at a distance of $c$ from the center. For a vertical hyperbola, the foci are at $(h, k \pm c)$, and for a horizontal hyperbola, the foci are at $(h \pm c, k)$.
Simplifying Square Roots: When simplifying square roots, it's often helpful to factor the number under the radical into its prime factors and then simplify by taking out pairs of prime factors as single numbers.
