Find the Vertex (x^2)/16+(y^2)/9=1
The question is asking to determine the vertex of the ellipse described by the equation (x^2)/16 + (y^2)/9 = 1. The vertex in the context of an ellipse typically refers to the points on the ellipse that are furthest from the center in the direction of the major axis. An ellipse has two pairs of vertices: one pair for the longer axis (major axis) and one pair for the shorter axis (minor axis). The given equation is in standard form, where the ellipse is centered at the origin (0,0) and the lengths of the semi-major and semi-minor axes are given by the denominators under the x^2 and y^2 terms, respectively.
$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$
Answer
Solution:
Step 1:
Rewrite the given equation to match the standard form of an ellipse equation, which is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, where the right side is already set to $1$.
$$\frac{x^2}{16} + \frac{y^2}{9} = 1$$
Step 2:
Recognize that the given equation is that of an ellipse. The standard form will help identify the center and the lengths of the axes.
$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$
Step 3:
Compare the given equation with the standard form to find the values of $a$, $b$, $h$, and $k$.
$$a = 4, b = 3, h = 0, k = 0$$
Step 4:
The center of the ellipse is given by the coordinates $(h, k)$. Insert the values for $h$ and $k$.
$$(0, 0)$$
Step 5:
Calculate the distance $c$ from the center to a focus of the ellipse.
Step 5.1:
Use the formula $c = \sqrt{a^2 - b^2}$ to determine $c$.
Step 5.2:
Plug in the values for $a$ and $b$.
$$c = \sqrt{4^2 - 3^2}$$
Step 5.3:
Perform the calculations.
Step 5.3.1:
Square the value of $4$.
$$\sqrt{16 - 3^2}$$
Step 5.3.2:
Square the value of $3$.
$$\sqrt{16 - 9}$$
Step 5.3.3:
Subtract $9$ from $16$.
$$\sqrt{7}$$
Step 6:
Determine the vertices of the ellipse.
Step 6.1:
Find the first vertex by adding $a$ to $h$.
$$(h + a, k)$$
Step 6.2:
Insert the known values for $h$, $a$, and $k$.
$$(0 + 4, 0)$$
Step 6.3:
Simplify the expression.
$$(4, 0)$$
Step 6.4:
Find the second vertex by subtracting $a$ from $h$.
$$(h - a, k)$$
Step 6.5:
Insert the known values for $h$, $a$, and $k$.
$$(0 - 4, 0)$$
Step 6.6:
Simplify the expression.
$$(-4, 0)$$
Step 6.7:
Note that an ellipse has two vertices.
$$(\text{Vertex})_1: (4, 0)$$ $$(\text{Vertex})_2: (-4, 0)$$
Step 7:
Locate the foci of the ellipse.
Step 7.1:
Find the first focus by adding $c$ to $h$.
$$(h + c, k)$$
Step 7.2:
Use the known values for $h$, $c$, and $k$.
$$(0 + \sqrt{7}, 0)$$
Step 7.3:
Simplify the expression.
$$(\sqrt{7}, 0)$$
Step 7.4:
Find the second focus by subtracting $c$ from $h$.
$$(h - c, k)$$
Step 7.5:
Use the known values for $h$, $c$, and $k$.
$$(0 - \sqrt{7}, 0)$$
Step 7.6:
Simplify the expression.
$$(-\sqrt{7}, 0)$$
Step 7.7:
Note that an ellipse has two foci.
$$(\text{Focus})_1: (\sqrt{7}, 0)$$ $$(\text{Focus})_2: (-\sqrt{7}, 0)$$
Step 8:
Calculate the eccentricity of the ellipse.
Step 8.1:
Use the formula for eccentricity $e = \frac{c}{a}$.
Step 8.2:
Substitute the values for $c$ and $a$.
$$e = \frac{\sqrt{7}}{4}$$
Step 8.3:
Simplify the expression to find the eccentricity.
$$e = \frac{\sqrt{7}}{4}$$
Step 9:
Summarize the important values for the ellipse.
- Center: $(0, 0)$
- Vertices: $(4, 0)$ and $(-4, 0)$
- Foci: $(\sqrt{7}, 0)$ and $(-\sqrt{7}, 0)$
- Eccentricity: $\frac{\sqrt{7}}{4}$
Step 10:
The solution is complete.
Knowledge Notes:
To solve for the vertex of an ellipse given in the standard form, it's important to understand the following concepts:
Standard Form of an Ellipse: The equation $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ represents an ellipse centered at the point $(h, k)$, where $a$ is the semi-major axis and $b$ is the semi-minor axis.
Vertices of an Ellipse: The vertices are points on the ellipse that lie along the major axis. They are found at $(h \pm a, k)$ and $(h, k \pm b)$, depending on whether the major axis is horizontal or vertical.
Foci of an Ellipse: The foci are two fixed points inside the ellipse such that the sum of the distances from any point on the ellipse to the foci is constant. They are found at $(h \pm c, k)$ and $(h, k \pm c)$, where $c = \sqrt{a^2 - b^2}$.
Eccentricity: The eccentricity of an ellipse, denoted by $e$, is a measure of how much the ellipse deviates from being circular. It is calculated using the formula $e = \frac{c}{a}$. For an ellipse, $0 < e < 1$.
Graphing an Ellipse: To graph an ellipse, plot the center, vertices, and foci, then draw a smooth curve connecting the vertices that encloses the foci.
Simplifying Square Roots: When simplifying expressions under a square root, remember to perform operations like squaring numbers and subtracting before taking the square root.
Understanding these concepts allows you to analyze and graph an ellipse, as well as find its key features such as the center, vertices, foci, and eccentricity.
