Find the Foci (x^2)/49+(y^2)/24=1
The question asks for the calculation of the coordinates of the foci of an ellipse described by the given equation (x^2)/49 + (y^2)/24 = 1. The foci of an ellipse are two points located along the major axis of the ellipse at equal distances from the center. The general equation provided is in standard form and represents an ellipse centered at the origin (0,0) with its major and minor axes aligned with the x and y axes, respectively. The task is to use the lengths of the major and minor axes determined from the given equation to find the exact positions of these foci according to the formula for the foci of an ellipse.
$\frac{x^{2}}{49} + \frac{y^{2}}{24} = 1$
Answer
Solution:
Step 1:
Rewrite the given equation so that it equals $1$ on the right-hand side. This is necessary for the equation to match the standard form of an ellipse or hyperbola. The equation is already in the desired form: $\frac{x^2}{49} + \frac{y^2}{24} = 1$.
Step 2:
Recognize that the equation represents an ellipse. The standard equation for an ellipse is $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$, which will help identify the ellipse's parameters.
Step 3:
Compare the given equation to the standard form to find the values for $a$, $b$, $h$, and $k$. Here, $a$ is the semi-major axis, $b$ is the semi-minor axis, and $(h, k)$ is the center of the ellipse. For our equation, we have $a = 7$, $b = 2\sqrt{6}$, $h = 0$, and $k = 0$.
Step 4:
Determine the distance $c$ from the center to a focus of the ellipse.
Step 4.1:
Use the formula $c = \sqrt{a^2 - b^2}$ to calculate the distance.
Step 4.2:
Plug the values of $a$ and $b$ into the formula: $c = \sqrt{7^2 - (2\sqrt{6})^2}$.
Step 4.3:
Perform the simplification.
Step 4.3.1:
Square the number $7$: $c = \sqrt{49 - (2\sqrt{6})^2}$.
Step 4.3.2:
Square $2\sqrt{6}$: $c = \sqrt{49 - 4 \cdot 6}$.
Step 4.3.3:
Subtract $4 \cdot 6$ from $49$: $c = \sqrt{49 - 24}$.
Step 4.3.4:
Complete the subtraction: $c = \sqrt{25}$.
Step 4.3.5:
Recognize that $\sqrt{25}$ is $5$: $c = 5$.
Step 5:
Locate the foci of the ellipse.
Step 5.1:
To find the first focus, add $c$ to $h$: $(h + c, k)$.
Step 5.2:
Insert the known values for $h$, $c$, and $k$: $(0 + 5, 0)$.
Step 5.3:
The result is the first focus: $(5, 0)$.
Step 5.4:
To find the second focus, subtract $c$ from $h$: $(h - c, k)$.
Step 5.5:
Insert the known values for $h$, $c$, and $k$: $(0 - 5, 0)$.
Step 5.6:
The result is the second focus: $(-5, 0)$.
Step 5.7:
An ellipse has two foci, which are located at $(\text{Focus})_1: (5, 0)$ and $(\text{Focus})_2: (-5, 0)$.
Step 6:
The foci of the ellipse given by the equation $\frac{x^2}{49} + \frac{y^2}{24} = 1$ are at $(5, 0)$ and $(-5, 0)$.
Knowledge Notes:
Ellipse Equation: The standard form of an ellipse is $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$, where $(h, k)$ is the center, $a$ is the semi-major axis, and $b$ is the semi-minor axis. If $a > b$, the ellipse is stretched along the x-axis, and if $b > a$, it is stretched along the y-axis.
Foci of an Ellipse: The foci of an ellipse are two fixed points located along the major axis, equidistant from the center. The distance from the center to each focus, denoted by $c$, is calculated using the formula $c = \sqrt{a^2 - b^2}$. The foci are important in the definition of an ellipse as the set of points for which the sum of the distances to the foci is constant.
Simplifying Square Roots: When simplifying expressions involving square roots, remember that $\sqrt{a^2} = a$ for any non-negative real number $a$.
Algebraic Manipulation: The steps involved in simplifying algebraic expressions include squaring numbers, applying the distributive property, and combining like terms.
Coordinate System: In a Cartesian coordinate system, points are denoted by $(x, y)$, where $x$ is the horizontal coordinate and $y$ is the vertical coordinate. The foci of an ellipse are expressed in this coordinate system.
