Find the Foci (y^2)/(96^2)-(x^2)/(40^2)=1
The question is asking you to determine the coordinates of the foci for an ellipse that is described by the given equation. An ellipse is a geometric figure that can be characterized by its major and minor axes, and its foci are two specific points along the major axis equidistant from the center. The equation provided is the standard form of an ellipse that is aligned with the coordinate axes and centered at the origin, with the y-axis being the major axis and the x-axis being the minor axis, since the y-term's denominator is larger. To find the foci, you would need to use the relationship between the distances of the vertices from the center and the focal distance from the center, typically using the equation c^2 = a^2 - b^2, where c is the focal distance, a is the semi-major axis, and b is the semi-minor axis.
$\frac{y^{2}}{\left(96\right)^{2}} - \frac{x^{2}}{\left(40\right)^{2}} = 1$
Answer
Solution:
Step 1: Standardize the Equation
The equation needs to be in the standard form for a hyperbola, which is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ for a vertical hyperbola or $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ for a horizontal hyperbola, where $1$ is on the right-hand side. In this case, the equation is already in the correct form, with $a^2 = 96^2$ and $b^2 = 40^2$:
$$\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1$$
Step 2: Identify the Values of $a$ and $b$
From the standard form, we can identify $a^2 = 96^2$ and $b^2 = 40^2$. Therefore, $a = 96$ and $b = 40$.
Step 3: Calculate the Distance of the Foci from the Center
For a hyperbola, the distance $c$ from the center to a focus is found using the formula $c^2 = a^2 + b^2$. Plugging in the values of $a$ and $b$, we get:
$$c^2 = 96^2 + 40^2$$ $$c^2 = 9216 + 1600$$ $$c^2 = 10816$$ $$c = \sqrt{10816}$$ $$c = 104$$
Step 4: Determine the Coordinates of the Foci
Since this is a vertical hyperbola (because the $y^2$ term comes first and has a positive coefficient), the foci are located at $(0, \pm c)$ along the y-axis. Therefore, the coordinates of the foci are:
$$(0, -104) \text{ and } (0, 104)$$
Knowledge Notes:
Hyperbola: A hyperbola is a type of conic section that is formed by the intersection of a right circular conical surface and a plane that cuts through both halves of the cone. It consists of two separate curves called branches.
Standard Form of a Hyperbola: The standard form of a hyperbola with a vertical transverse axis is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, and with a horizontal transverse axis is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $a$ 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.
Foci of a Hyperbola: The foci (plural of focus) are two fixed points located inside each branch of the hyperbola that have the property that the difference in the distances from any point on the hyperbola to the foci is constant. The foci are always located along the transverse axis.
Distance to the Foci: The distance $c$ from the center of the hyperbola to each focus is calculated using the formula $c^2 = a^2 + b^2$. This is derived from the definition of a hyperbola and the Pythagorean theorem.
Coordinates of the Foci: For a vertical hyperbola, the foci are at $(h, k \pm c)$, and for a horizontal hyperbola, the foci are at $(h \pm c, k)$, where $(h, k)$ is the center of the hyperbola. In the given problem, the center is at the origin $(0, 0)$.
Understanding these concepts is crucial for solving problems related to hyperbolas, including finding their foci, vertices, and asymptotes, as well as graphing them.
