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Ellipses Solutions

Ellipses are similar to circles in that they are both closed curves. However, while all circles are ellipses, not all ellipses are circles. 
An ellipse is the set of all points in a plane where the sum of the distances from two fixed points is constant . 
The fixed points are called foci . (One is called a focal point.)

Note

Note: To learn more about the foci of an ellipse, complete the More to Explore.

The standard form of an ellipse is: $\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1$  
(h, k) represents the center point. 
a, b represent half the length of the axes (where a is horizontal and b is vertical) and must be > 0 . 
Of the two axes, a or b, one is major (longer) and the other is minor (shorter), the combination of which gives the ellipse its squished or stretched shape. 
The value of a and b determines: 
- which of the axes is the major axis, and 
- if the ellipse will be horizontal or vertical . 
Vertices are where the ellipse intersects the major axis. 
Co-vertices are where the ellipse intersects the minor axis. 
Vertices and co-vertices are sometimes referred to as endpoints because they determine the values of the domain and range.

Horizontal Major Axis

a > b
Center: (h, k)
Vertices: $(h - a, k)$ and $(h + a, k)$ or $(h \pm a, k)$
Co-Vertices: $(h, k - b)$ and $(h, k + b)$ or $(h, k \pm b)$

Vertical Major Axis

a < b
Center: (h, k)
Vertices: $(h, k - b)$ and $(h, k + b)$ or $(h, k \pm b)$
Co-Vertices: $(h - a, k)$ and $(h + a, k)$ or $(h \pm a, k)$

When asked to write the equation of an ellipse from a graph you need the center , and the values of a and b .

Note

It is possible to graph an ellipse along a slant axis (not aligned with either the x‐ or y-axis). Ellipses with slant axes are not covered in this level.

Example 1

Write the equation of the ellipse on the coordinate plane. Name the length of the major axis and minor axis.

Plan
Determine major axis  

Mark on graph: center, vertices, co-vertices 

Write equation 

Name the length of each axis

Horizontal major axis: $a > b$  

Center: (–4, 3) 

$\begin{matrix} a = 8 & b = 5 \\ a^{2} = 64 & b^{2} = 25 \end{matrix}$

  $\frac{{(x + 4)}^{2}}{64} + \frac{{(y - 3)}^{2}}{25} = 1$  

Major axis = 16  

Minor axis = 10

Note

You can mark a and b on the graph and use mental math to determine $a^{2}$ and $b^{2}$ rather than showing the work.

Example 2

Write the equation of ellipse K and ellipse N. Name the co-vertices of ellipse K and the vertices of ellipse N.

Ellipse K 

Horizontal major axis: $a > b$  

Center: (–4, 3) 

$\begin{array}{rcl} \begin{matrix} a = 5 & b = 3 \\ a^{2} = 25 & b^{2} = 9 \end{matrix} \end{array}$  

$\frac{{(x + 4)}^{2}}{25} + \frac{{(y - 3)}^{2}}{9} = 1$  

Co-vertices: $(- 4, 6), (- 4, 0)$

Ellipse N 

Vertical major axis: $b > a$  

Center: $(5, - 1)$

  $\begin{array}{rcl} \begin{matrix} a = 8 & b = 9 \\ a^{2} = 64 & b^{2} = 81 \end{matrix} \end{array}$

  $\frac{{(x - 5)}^{2}}{64} + \frac{{(y + 1)}^{2}}{81} = 1$  

Vertices: $(5, 8), (5, - 10)$