The eccentricity of a conic section is a measure of how much the shape deviates from a circle.

If two conic sections have the same eccentricity, then they are similar.

Notes

\(\text{Eccentricity}=\displaystyle\frac{\text{Distance from Center to Focus (c)}}{\text{Distance from Center to Vertex (a)}}\)

\({\text{Conic Sections & Eccentricity}}\)
\(\underline{\text{Conic Section}}\)	\(\underline{\text{Eccentricity (e)}}\)
\(\text{Circle}\)	\(e=0\)
\(\text{Ellipse}\)	\(0 \lt e \lt 1\)
\(\text{Parabola}\)	\(e=1\)
\(\text{Hyperbola}\)	\(e \gt 1\)

\({\text{Ellipses}}\)
\(\text{Equation}\)	\(\displaystyle\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\) \(a\gt b\)	\(\displaystyle\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1\) \(a\gt b\)
\(\text{Shape}\)
\(\text{Foci}\)	\((h \pm c,k)\)	\((h ,k \pm c)\)
\(\text{Vertices}\)	\((h \pm a,k)\)	\((h ,k \pm a)\)
\(\text{Center}\)	\((h,k)\)
\(\text{Focal Length Equation}\)	\(a^2-b^2=c^2\)
\(\text{Length of Major Axis}\)	\(2a\)
\(\text{Length of Minor Axis}\)	\(2b\)
\(\text{Eccentricity}\)	\(\displaystyle\frac{c}{a}\)

Practice Problems

\(\textbf{1)}\) Find the eccentricity of the following circle.
\(\displaystyle(x-4)^2+y^2=16\)

Show Eccentricity

The eccentricity is \( e=0 \)

Show Work

All Circles have eccentricity \( e=0 \)

\(\textbf{2)}\) Find the eccentricity of the following ellipse.
\(\displaystyle\frac{x^2}{16}+\displaystyle\frac{y^2}{25}=1\)

Show Eccentricity

\(\,\,\,\)The eccentricity is \( e=\displaystyle\frac{3}{5} \)

Show Work

\(\,\,\,\text{For ellipses, } a^2-b^2=c^2\)

\(\,\,\, 25-16=c^2\)

\(\,\,\, 9=c^2\)

\(\,\,\, c=3\)

\(\,\,\, e=\displaystyle \frac{c}{a}\)

\(\,\,\, e=\displaystyle \frac{3}{5}\)

\(\textbf{3)}\) Find the eccentricity of the following parabola.
\(\displaystyle y= 4(x+2)^2+3\)

Show Eccentricity

The eccentricity is \( e=1 \)

Show Work

All Parabolas have eccentricity \( e=1 \)

\(\textbf{4)}\) Find the eccentricity of the following hyperbola.
\(\displaystyle\frac{x^2}{144}-\displaystyle\frac{y^2}{25}=1\)

Show Eccentricity

\(\,\,\,\)The eccentricity is \( e=\displaystyle\frac{13}{12} \)

Show Work

\(\,\,\,\text{For hyperbolas, } a^2+b^2=c^2\)

\(\,\,\, 144+25=c^2\)

\(\,\,\, 169=c^2\)

\(\,\,\, c=13\)

\(\,\,\, e=\displaystyle \frac{c}{a}\)

\(\,\,\, e=\displaystyle \frac{13}{12}\)

Challenge Problem

\(\textbf{5)}\) Find the equation of the ellipse with vertices at (-10, 0) and (10, 0) with an eccentricity of \(\frac{3}{5}\).

Show Equation

\(\,\,\,\)The equation is \(\displaystyle\frac{x^2}{100}+\frac{y^2}{64}=1 \)

Show Work

\(\,\,\,\left(\text{Center is Midpoint of Vertices}\right)\,\,\, \displaystyle\left(\frac{-10+10}{2},\frac{0-0}{2}\right)=\left(0,0\right)\)

\(\,\,\,\left(a= \text{Distance from Center to a Vertex}\right)\,\,\, a=10\)

\(\,\,\,\left(\text{Eccentricity Formula}\right)\,\,\, \displaystyle e=\frac{c}{a}\)

\(\,\,\,\left(\text{Fill in } e \text{ and } a\right),\,\,\, \displaystyle \frac{3}{5}=\frac{c}{10}\)

\(\,\,\,\left(\text{Cross Multiply}\right)\,\,\, \displaystyle 30=5c\)

\(\,\,\, c=6\)

\(\,\,\,\left(\text{For ellipses}\right)\,\,\, a^2-b^2=c^2\)

\(\,\,\, 100-b^2=36\)

\(\,\,\, b^2=64\)

\(\,\,\, b=8\)

\(\,\,\,\left(\text{Equation Horizontal Ellipses}\right)\,\,\, \displaystyle\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1\)

\(\,\,\,\left(\text{Fill in what we know.}\right)\,\,\, \displaystyle\frac{(x-0)^2}{100}+\frac{(y-0)^2}{64}=1\)

\(\,\,\,\left(\text{Clean it up}\right) \displaystyle\frac{x^2}{100}+\frac{y^2}{64}=1\)

See Related Pages\(\)

\(\bullet\text{ All Conic Section Notes}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Equation of a Circle}\)
\(\,\,\,\,\,\,\,\,(x-h)^2+(y-k)^2=r^2…\)

\(\bullet\text{ Parabolas}\)
\(\,\,\,\,\,\,\,\,y=a(x-h)^2+k…\)

\(\bullet\text{ Axis of Symmetry}\)
\(\,\,\,\,\,\,\,\,x=-\frac{b}{2a}…\)

\(\bullet\text{ Ellipses}\)
\(\,\,\,\,\,\,\,\,\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1…\)

\(\bullet\text{ Area of Ellipses}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\pi a b…\)

\(\bullet\text{ Hyperbolas}\)
\(\,\,\,\,\,\,\,\,\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1…\)

\(\bullet\text{ Conic Sections- Completing the Square}\)
\(\,\,\,\,\,\,\,\,x^2+8x+y^2−6y=11 \Rightarrow (x+4)^2+(y−3)^2=36…\)

\(\bullet\text{ Conic Sections- Parametric Equations}\)
\(\,\,\,\,\,\,\,\,x=h+r \cos{t}\)
\(\,\,\,\,\,\,\,\,y=k+r \sin{t}…\)

\(\bullet\text{ Degenerate Conics}\)
\(\,\,\,\,\,\,\,\,x^2−y^2=0…\)

\(\bullet\text{ Andymath Homepage}\)

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