12. Conic Sections & Systems of Nonlinear Equations

Ellipses

12. Conic Sections & Systems of Nonlinear Equations

Ellipses: Videos & Practice Problems

Topic summary

An ellipse is an oval shape defined by points where the sum of distances to two foci is constant, contrasting with a circle's single-radius distance. Its equation resembles a circle's but includes two distances: $x^{2} / a^{2} + y^{2} / b^{2} = 1$ , where $a$ and $b$ represent horizontal and vertical radii. Graphing involves plotting points $a$ units left/right and $b$ units up/down from the center, then connecting smoothly, highlighting the ellipse's relation to polynomials and geometric transformations.

concept

Ellipses At The Origin

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Ellipses At The Origin Video Summary

An ellipse is a geometric shape closely related to a circle but with distinct properties. While a circle consists of all points equidistant from a single center point, an ellipse is defined as the set of all points where the sum of the distances to two fixed points, called the foci, remains constant. This difference means that instead of a single radius, an ellipse has two key distances that determine its shape.

Ellipses can be oriented either vertically or horizontally. In a vertically stretched ellipse, the foci lie along the y-axis, and the ellipse extends further in the vertical direction. Conversely, in a horizontally stretched ellipse, the foci are positioned along the x-axis, and the ellipse stretches more horizontally. The two main parameters that describe an ellipse are a and b, representing the distances from the center to the ellipse along the major and minor axes, respectively. For a vertical ellipse, b is the longer distance (vertical stretch), and a is the shorter (horizontal). For a horizontal ellipse, a is the longer distance (horizontal stretch), and b is the shorter (vertical).

The equation of an ellipse centered at the origin is similar in form to the equation of a circle but accounts for these two distances. The general form is:

\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]

Here, a and b are the lengths of the semi-major and semi-minor axes, respectively. For example, if an ellipse is vertically stretched with a = 3 and b = 4, its equation becomes:

\[\frac{x^2}{3^2} + \frac{y^2}{4^2} = 1 \quad \Rightarrow \quad \frac{x^2}{9} + \frac{y^2}{16} = 1\]

For a horizontally stretched ellipse with a = 4 and b = 3, the equation is:

\[\frac{x^2}{4^2} + \frac{y^2}{3^2} = 1 \quad \Rightarrow \quad \frac{x^2}{16} + \frac{y^2}{9} = 1\]

Graphing an ellipse involves plotting points at distances a units left and right, and b units up and down from the center, typically the origin. Connecting these points with a smooth, oval curve forms the ellipse. This method parallels graphing a circle but requires attention to the two different radii.

Understanding ellipses deepens comprehension of conic sections and their properties, highlighting how stretching a circle along one axis transforms it into an ellipse. Recognizing the role of the foci and the relationship between a and b is essential for analyzing and graphing ellipses effectively.

Problem

Write the standard form equation of each ellipse centered at the origin.

$\frac{x^{2}}{4} + y^{2} = 1$

$x^{2} + \frac{y^{2}}{16} = 1$

$x^{2} + y^{2} = 16$

$\frac{x^2}{16}+y^2=1$

Problem

Write the standard form equation of each ellipse centered at the origin.

$\frac{x^2}{4}+\frac{y^2}{25}=1$

$\frac{x^2}{25}+\frac{y^2}{4}=1$

$\frac{x^2}{2}+\frac{y^2}{5}=1$

$\frac{x^2}{5}+\frac{y^2}{2}=1$

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Ellipses At The Origin Example 1

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Ellipses At The Origin Example 1 Video Summary

To find the x- and y-intercepts of an ellipse given by an equation, start by recalling that the x-intercepts occur where y = 0, and the y-intercepts occur where x = 0. For the ellipse equation, setting y = 0 simplifies the equation, allowing you to solve for x. For example, if the equation is of the form $\frac{x^2}{16} + \frac{y^2}{25} = 1$, substituting y = 0 reduces it to $\frac{x^2}{16} = 1$. Multiplying both sides by 16 gives $x^2 = 16$, and taking the square root yields $x = \pm 4$. Thus, the x-intercepts are at (4, 0) and (-4, 0).

Similarly, to find the y-intercepts, set x = 0. This simplifies the equation to $\frac{y^2}{25} = 1$. Multiplying both sides by 25 results in $y^2 = 25$, and taking the square root gives $y = \pm 5$. Therefore, the y-intercepts are at (0, 5) and (0, -5).

Knowing these intercepts and that the ellipse is centered at the origin, you can plot the points accordingly: four units left and right along the x-axis, and five units up and down along the y-axis. Connecting these points smoothly forms the characteristic oval shape of the ellipse. This process highlights the importance of understanding how to manipulate and interpret the standard form of an ellipse equation to identify key features such as intercepts and center, which are essential for accurate graphing.

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Ellipses At The Origin Example 2

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Ellipses At The Origin Example 2 Video Summary

To graph an ellipse given by an equation that is not initially in the standard form, it is essential to first rewrite the equation to match the standard ellipse equation centered at the origin:

\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]

When the equation has a constant other than 1 on the right side, such as 36, divide both sides of the equation by that constant to normalize it. For example, dividing both sides by 36 transforms the equation into a form where the right side equals 1. This step is crucial because the standard form requires the right side to be 1 for easy identification of the ellipse's parameters.

Next, simplify the coefficients of the squared terms. If the coefficients are products of perfect squares, factor and reduce them accordingly. For instance, if the coefficients are 36, which equals 4 times 9, dividing through by 36 allows cancellation of these factors, resulting in terms like \[\frac{x^2}{9} + \frac{y^2}{4} = 1\].

From this form, identify the values of \[a^2\] and \[b^2\] directly from the denominators. Here, \[a^2 = 9\] and \[b^2 = 4\]. Taking the square roots gives the lengths of the semi-major and semi-minor axes: \[a = 3\] and \[b = 2\].

Since the ellipse is centered at the origin, plot points at distances \[a\] units left and right along the x-axis, and \[b\] units up and down along the y-axis. Specifically, mark points at (3, 0), (-3, 0), (0, 2), and (0, -2). Connecting these points smoothly forms the characteristic oval shape of the ellipse.

This method highlights the importance of manipulating the given equation to fit the standard ellipse form, enabling straightforward identification of key parameters and accurate graphing. By dividing through by the constant on the right side and simplifying coefficients, any ellipse equation can be transformed for easy visualization and analysis.

Here’s what students ask on this topic:

An ellipse is an oval-shaped curve defined as the set of all points where the sum of the distances to two fixed points, called foci, is constant. This contrasts with a circle, which is the set of points equidistant from a single center point. In a circle, every point on the edge is the same distance (radius) from the center. In an ellipse, the sum of distances from any point on the curve to the two foci remains constant, but the individual distances vary. This difference results in the ellipse having two axes of symmetry: a longer axis (major axis) and a shorter axis (minor axis), whereas a circle has equal radii in all directions.

The standard equation of an ellipse centered at the origin is given by $x^{2} / a^{2} + y^{2} / b^{2} = 1$ . Here, $a$ and $b$ represent the lengths of the semi-major and semi-minor axes, respectively. If $a$ is greater than $b$ , the ellipse is stretched horizontally; if $b$ is greater, it is stretched vertically. This equation generalizes the circle equation $x^{2} + y^{2} = r^{2}$ by allowing different radii along the x and y axes.

To graph an ellipse given its equation in standard form $x^{2} / a^{2} + y^{2} / b^{2} = 1$ , first identify the values of $a$ and $b$ . These represent the distances from the center to the ellipse along the x-axis and y-axis, respectively. Starting at the center (usually the origin), plot points $a$ units to the right and left, and $b$ units up and down. Then, connect these points smoothly to form the oval shape of the ellipse. This method works for both horizontal and vertical ellipses, depending on which axis has the larger radius.

The foci (singular: focus) of an ellipse are two fixed points located along the major axis such that the sum of the distances from any point on the ellipse to these two foci is constant. The positions of the foci depend on the values of $a$ and $b$ in the ellipse equation. The distance from the center to each focus, denoted as $c$ , is found using the relationship $c^{2} = a^{2} - b^{2}$ . The foci lie along the major axis at positions $(\pm c, 0)$ for a horizontal ellipse or $(0, \pm c)$ for a vertical ellipse. Understanding the foci is essential for grasping the geometric definition of an ellipse.

You can determine the orientation of an ellipse by comparing the values of $a$ and $b$ in its standard equation $x^{2} / a^{2} + y^{2} / b^{2} = 1$ . If $a$ (the denominator under $x^{2}$ ) is greater than $b$ , the ellipse is stretched horizontally, meaning the major axis lies along the x-axis. Conversely, if $b$ (the denominator under $y^{2}$ ) is greater, the ellipse is stretched vertically, with the major axis along the y-axis. This distinction helps in graphing and understanding the ellipse's shape.