Find the Foci ((x+1)^2)/225-((y+5)^2)/400=1
The given problem is to determine the locations of the foci of an ellipse represented by the equation ((x+1)^2)/225 - ((y+5)^2)/400 = 1. The equation is provided in the standard form of an ellipse with its center not at the origin, where the terms represent the squared distances from the center to the vertices along the x-axis and y-axis, respectively. The question is asking for the specific points along the major axis of the ellipse where the foci are situated. These points are unique to an ellipse and play a key role in its geometric definition.
$\frac{\left(\left(\right. x + 1 \left.\right)\right)^{2}}{225} - \frac{\left(\left(\right. y + 5 \left.\right)\right)^{2}}{400} = 1$
Answer
Solution:
Step 1:
Rewrite the given equation to conform to the standard form of a hyperbola, where the right side equals $1$.
$$\frac{(x + 1)^2}{225} - \frac{(y + 5)^2}{400} = 1$$
Step 2:
Identify the equation as a hyperbola by comparing it with the general equation of a hyperbola.
$$\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$$
Step 3:
Determine the values corresponding to $h$, $k$, $a$, and $b$ by matching them with the standard form.
$$a = 15$$ $$b = 20$$ $$k = -5$$ $$h = -1$$
Step 4:
Calculate the distance $c$ from the center to a focus of the hyperbola.
Step 4.1:
Use the equation to find $c$.
$$c = \sqrt{a^2 + b^2}$$
Step 4.2:
Insert the known values for $a$ and $b$.
$$c = \sqrt{15^2 + 20^2}$$
Step 4.3:
Perform the arithmetic operations.
Step 4.3.1:
Square $15$.
$$c = \sqrt{225 + 20^2}$$
Step 4.3.2:
Square $20$.
$$c = \sqrt{225 + 400}$$
Step 4.3.3:
Add the squared terms.
$$c = \sqrt{625}$$
Step 4.3.4:
Express $625$ as a square of an integer.
$$c = \sqrt{25^2}$$
Step 4.3.5:
Extract the square root.
$$c = 25$$
Step 5:
Locate the foci of the hyperbola.
Step 5.1:
To find the first focus, add $c$ to $h$.
$$(h + c, k)$$
Step 5.2:
Plug in the values for $h$, $c$, and $k$.
$$(24, -5)$$
Step 5.3:
To find the second focus, subtract $c$ from $h$.
$$(h - c, k)$$
Step 5.4:
Insert the known values for $h$, $c$, and $k$.
$$(-26, -5)$$
Step 5.5:
The foci of the hyperbola are given by the general formula $(h \pm c, k)$.
$$(24, -5), (-26, -5)$$
Step 6:
The foci of the hyperbola are located at $(24, -5)$ and $(-26, -5)$.
Knowledge Notes:
A hyperbola is a type of conic section that appears as two separate curves, each one being a mirror image of the other. It can be defined as the set of all points in the plane where the difference in distances from two fixed points (foci) is constant.
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 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 a focus is found using the formula $c = \sqrt{a^2 + b^2}$ for hyperbolas.
The foci of a hyperbola are located at $(h \pm c, k)$ for a horizontal hyperbola and $(h, k \pm c)$ for a vertical hyperbola.
The vertices of a hyperbola are points where the hyperbola intersects its transverse axis, and they are located at $(h \pm a, k)$ for a horizontal hyperbola and $(h, k \pm a)$ for a vertical hyperbola.
Asymptotes of a hyperbola are straight lines that the hyperbola approaches but never touches. They pass through the center of the hyperbola and have slopes of $\pm \frac{b}{a}$ for a horizontal hyperbola and $\pm \frac{a}{b}$ for a vertical hyperbola.
