Ellipses: Standard Form: Videos & Practice Problems

In the study of conic sections, the ellipse is a fascinating shape that emerges when a three-dimensional cone is intersected by a two-dimensional plane at a slight angle. Unlike the circle, which is defined by a single radius, the ellipse requires the consideration of two distinct distances: the semi-major axis and the semi-minor axis. The semi-major axis, denoted as \( a \), represents the longest distance from the center of the ellipse to its edge, while the semi-minor axis, denoted as \( b \), is the shorter distance.

When visualizing a horizontal ellipse, the semi-major axis extends along the x-axis, and the semi-minor axis extends along the y-axis. The equation for a horizontal ellipse can be expressed as:

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

For example, if \( a = 4 \) and \( b = 3 \), the equation becomes:

\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]

Conversely, when the ellipse is vertically stretched, the semi-major axis aligns with the y-axis, and the semi-minor axis aligns with the x-axis. The equation for a vertical ellipse is structured as follows:

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

Using the same values, the equation for a vertical ellipse would be:

\[ \frac{x^2}{9} + \frac{y^2}{16} = 1 \]

It is essential to recognize that the variable \( a \) consistently indicates the longest distance from the center to the ellipse's edge, regardless of the orientation of the ellipse. This distinction is crucial for accurately graphing and understanding the properties of ellipses.

Interestingly, the equations of ellipses share a resemblance with the equation of a circle. The standard equation of a circle is given by:

\[ x^2 + y^2 = r^2 \]

By dividing through by \( r^2 \), it can be rewritten as:

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

This form highlights the symmetry of the circle, where the radius is uniform in all directions. In contrast, the ellipse equations reflect the varying distances \( a \) and \( b \), indicating the lack of uniform symmetry.

Understanding these concepts lays the groundwork for further exploration of ellipses, including their properties, applications, and the relationships they share with other conic sections.

In the study of ellipses, understanding the concepts of vertices and foci is essential for graphing and analyzing their properties. An ellipse has two vertices and two foci, which are critical points located along the major axis of the ellipse. The vertices are the points that are furthest from the center of the ellipse, while the foci are points that help define the ellipse's symmetry.

For a horizontally stretched ellipse, the vertices are positioned along the major axis, which is horizontal. To find the vertices, you need the distance a, known as the semi-major axis. The vertices can be calculated by moving a units to the right and left from the center, resulting in coordinates at (a, 0) and (-a, 0).

The foci, on the other hand, are determined using the relationship between the semi-major axis and the semi-minor axis. The distance to the foci, denoted as c, can be calculated using the equation:

\[c^2 = a^2 - b^2\]

where b is the semi-minor axis. The foci are then located at (c, 0) and (-c, 0) for a horizontal ellipse.

For example, if the equation of an ellipse is given and the largest denominator is 25, then a is calculated as:

\[a = \sqrt{25} = 5\]

Thus, the vertices would be at (5, 0) and (-5, 0). If the smaller denominator is 9, then:

\[b = \sqrt{9} = 3\]

Using the equation for c:

\[c^2 = 25 - 9 = 16 \implies c = \sqrt{16} = 4\]

Therefore, the foci would be at (4, 0) and (-4, 0).

In contrast, for a vertically stretched ellipse, the vertices and foci are located along the y-axis. The vertices would be at (0, a) and (0, -a), while the foci would be at (0, c) and (0, -c). This distinction is crucial when graphing ellipses and understanding their geometric properties.

In summary, the vertices and foci of an ellipse provide insight into its shape and symmetry, with their positions determined by the semi-major and semi-minor axes. Mastering these calculations is fundamental for effectively working with ellipses in mathematics.

When dealing with ellipses that are not centered at the origin, the equation takes the form \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\), where \(h\) and \(k\) represent the horizontal and vertical shifts, respectively. This transformation is similar to function transformations, where the graph shifts based on the values of \(h\) and \(k\).

To graph an ellipse, the first step is to identify the lengths of the major and minor axes. The major axis is associated with the larger denominator in the equation. For example, if the equation yields \(a^2 = 64\) and \(b^2 = 9\), then \(a\) (the semi-major axis) is \(\sqrt{64} = 8\) and \(b\) (the semi-minor axis) is \(\sqrt{9} = 3\).

Next, determine the orientation of the ellipse. If the larger number (in this case, \(64\)) is under the \(y\) term, the ellipse is vertical. Conversely, if it were under the \(x\) term, the ellipse would be horizontal.

The center of the ellipse is found using the values of \(h\) and \(k\) from the equation. For instance, if the equation is \((x-4)^2/9 + (y-2)^2/64 = 1\), then \(h = 4\) and \(k = 2\), placing the center at the point \((4, 2)\).

To find the vertices of a vertical ellipse, add and subtract the value of \(a\) from \(k\). This means moving up and down from the center. For example, from the center \((4, 2)\), moving up \(8\) units gives the vertex at \((4, 10)\), and moving down \(8\) units gives the vertex at \((4, -6)\).

For the \(b\) points, which are horizontal, you add and subtract \(b\) from \(h\). Starting from the center \((4, 2)\), moving left \(3\) units gives the point \((1, 2)\), and moving right \(3\) units gives the point \((7, 2)\).

Finally, to graph the ellipse, connect the vertices and \(b\) points with a smooth curve. Additionally, to find the foci of a vertical ellipse, calculate \(c\) using the formula \(c = \sqrt{a^2 - b^2}\). For the example given, \(c = \sqrt{64 - 9} = \sqrt{55}\). The foci will be located at \((h, k \pm c)\), which means you would add and subtract \(c\) from the vertical position \(k\).

Understanding these steps allows for the effective graphing and analysis of ellipses that are not centered at the origin, reinforcing the concepts of transformations and geometric properties.