Precalculus - Ellipses

Introduction

• Ellipse is the set of all points in a plane such the sum of the distances from two fixed points, called the foci, is constant. 
• An ellipse is formed when the cone is cut obliquely to the axis and the surface of the cone.
• To roughly visualize an ellipse, you can imagine two parabolas joined together. 
• Study the graph of the ellipse below.

• The graph of an ellipse crosses the major axis at two fixed points called vertices typically written in symbols $V_1 and V_2$. The ellipse above has vertices at $V_1\left(0,-2\right) and V_2\left(10,-2\right)$.
• The coordinates of the center are at $C\left(h,k\right)=C\left(5,-2\right)$.
• The major axis in the example above is the line segment $AB$ while the minor axis is the line segment $CD$. The minor axis is the line segment through the center perpendicular to the major axis.
• The length of the major axis is given by $d_1+d_2=2a$. In the case above, $d_1=AG and d_2=BG, so we have AG+BG=2a$.
• The coordinates of the foci are typically given by the symbols $F_1\left(h+c,k\right) and F_2\left(h-c,k\right)$ if the major axis is parallel to the x-axis. In the example above, the coordinates of the foci are $F_1\left(8,-2\right) and F_2\left(2,-2\right)$.
• The line segment through the coordinates of the foci is called the latus rectum (focal chord). The length of the latus rectum is given by $L=\frac{2b^2}{a}$.
• The eccentricity of the ellipse is $0<e<1$ or $e=\frac{c}{a}$.
• The general equation of the ellipse is given by $A{x}^2+C{y}^2+Dx+Ey+F=0 where A\ne C and AC>0$.
• The discriminant of an ellipse is $b^2-4ac<0 where ac>0$.
• The circumference of an ellipse is given by $C=2\mathrm{\pi }\left(\sqrt{\frac{{\mathrm{a}}^2+{\mathrm{b}}^2}{2}}\right)$ and the area of the ellipse is given by $A=\mathrm{\pi }\sqrt{\mathrm{AC}}$.

Forms and Graphs of Ellipse

1. Center at $C\left(0,0\right)$ with major axis = x-axis and minor axis = y-axis

Graph: 

Standard Equation is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 where a>b$. 

To find the value of a, b, or c, we use $a^2=b^2+c^2$.

The coordinates of the vertices are $V_1\left(a,0\right) and V_2\left(-a,0\right)$.

Co-vertices are $B_1\left(0,b\right) and B_2\left(0,-b\right)$.

The coordinates of the foci are $F_1\left(c,0\right) and F_2\left(-c,0\right)$.

The equation of the directrix is given by $x=\pm \frac{a}{e}$.

 

2. Center at $C\left(0,0\right)$ with major axis = y-axis and minor axis = x-axis.

Graph:

Standard Equation is $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ where $a>b$.

To find the value of a, b, or c, we use $a^2=b^2+c^2$.

The coordinates of the vertices are $V_1\left(0,a\right) and V_2\left(0,-a\right)$.

Co-vertices are $B_1\left(b,0\right) and B_2\left(-b,0\right)$.

The coordinates of the foci are $F_1\left(0,c\right) and F_2\left(0,-c\right)$.

The equation of the directrix is given by $y=\pm \frac{a}{e}$.

 

3. Center at $C\left(h,k\right)$, major axis is parallel to the x-axis, and the minor axis is parallel to the y-axis.

Graph:

The standard Equation is $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$ where $a>b$

To find the value of a, b, or c, we use $a^2=b^2+c^2$. 

The coordinates of the vertices are $V_1\left(h+a,k\right) and V_2\left(h-a,k\right)$.

Co-vertices are $B_1\left(h,k+b\right) and B_2\left(h,k-b\right)$

The coordinates of the foci are $F_1\left(h+c,k\right) and F_2\left(h-c,k\right)$

The equation of the directrix is $x=h\pm \frac{a^2}{c}$

 

4. Center at $C\left(h,k\right)$, the major axis is parallel to the y-axis, and the minor axis is parallel to the x-axis.

Graph:

Standard equation is $\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1 where a>b$

To find the value of a, b, or c, we use $a^2=b^2+c^2$.

The coordinates of the vertices are $V_1\left(h,k+a\right) and V_2\left(h,k-a\right)$.

Co-vertices are $B_1\left(h+b,k\right) and B_2\left(h-b,k\right)$

The coordinates of the foci are $F_1\left(h,k+c\right) and F_2\left(h,k-c\right)$

The equation of the directrix is $y=k\pm \frac{a^2}{c}$

Solved Examples

Example 1. Given the ellipse $\frac{{\left(x+9\right)}^2}{121}+\frac{{\left(y-3\right)}^2}{81}=1$, find the coordinates of the center, foci, and vertices. 

Solution:

The major axis is parallel to the x-axis.

Based on the given equation, we have $h=-9 and k=3$. Thus, the center is at $C\left(-9,3\right)$.

To solve for $a$, we look for the larger denominator and take its square root. Hence, $a=11$ and $b=9$.

To solve for $c$, we use $a^2=b^2+c^2$.

${\left(11\right)}^2={\left(9\right)}^2+c^2$

$c=2\sqrt{10}$

The coordinates of the foci are $F_1\left(-9+2\sqrt{10},3\right) and F_2\left(-9-2\sqrt{10},3\right)$.

The coordinates of the vertices are $B_1\left(-9,12\right) and B_2\left(-9,-6\right)$

 

Example 2. Find the equation of the ellipse with foci $\left(4,7\right) and \left(4,-9\right)$ and a major axis length of 34 units.

Solution:

By inspection of the given points, the ellipse has a major axis parallel to the y-axis. It follows the standard equation $\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1 where a>b$.

To solve for a, we have:

$2a=34$

$a=17$

To solve for c, we use the y-coordinates of the foci.

$\left\{\begin{array}{l}k-c=-9\\ k+c=7\end{array}\right.$

Solving the system yields $k=-1 and c=8$.

To solve for b, we use $a^2=b^2+c^2$.

${\left(17\right)}^2=b^2+{\left(8\right)}^2$

$b^2=225\to b=15$

Hence, the equation of the ellipse is $\frac{{\left(x-4\right)}^2}{225}+\frac{{\left(y+1\right)}^2}{289}=1$.

 

Example 3. Write the equation of the ellipse $x^2+2y^2-6x+20y+41=0$ in standard form.

Solution: 

Use completing the square. To illustrate, we have:

$x^2-6x+2y^2+20y=-41$

$x^2-6x+9+2\left(y^2+10y+25\right)=-41+9+2\left(25\right)$

${\left(x-3\right)}^2+2{\left(y+5\right)}^2=18\to \frac{{\left(x-3\right)}^2}{18}+\frac{{\left(y+5\right)}^2}{9}$

Cheat Sheet

• The ellipse $\frac{{\left(x-7\right)}^2}{12}+\frac{{\left(y-2\right)}^2}{8}=1$ has its center at $C\left(7,2\right)$ with a major axis parallel to the x-axis.
• The ellipse $\frac{{\left(x+{\displaystyle \frac{3}{4}}\right)}^2}{225}+\frac{{\left(y-2\right)}^2}{400}=1$ has its center at $C\left(-\frac{3}{4},2\right)$ with a major axis parallel to the y-axis.
• The ellipse in the form of $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$ has a horizontal major axis.
• The ellipse $\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1$ has a vertical major axis.
• Always be careful of the sign when dealing with coordinates of the center, foci, vertices, and co-vertices.
• Be mindful of the difference between $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1 and \frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1 where a>b$.

Blunder Areas

• Don't be confused with the equation $a^2=b^2+c^2$ and the Pythagorean Theorem $c^2=a^2+b^2$. 
• It is a common mistake to identify cases like $\frac{{\left(x+3\right)}^2}{18}+\frac{{\left(y+5\right)}^2}{20}=1$ has their center at $C\left(3,5\right)$. The correct coordinates of the center is $C\left(-3,-5\right)$.
• To avoid confusion in solving problems involving ellipses, always check or inspect the points given. The coordinates and location of the center, foci, vertices, and co-vertices determine the type of ellipse and its major axis. 
• Be careful in using the values of $a$ and $b$ for the equation $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$. It is by default that $a>b.$ The position of $a and b$ changes depending on the major axis.