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Lesson

Notes

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

Questions

Graph the following ellipses.

\(\textbf{1)}\) \(\displaystyle\frac{x^2}{4}+\displaystyle\frac{y^2}{25}=1\)

Show Hint

This is the same as \(\displaystyle\left(\frac{x}{2}\right)^2+\displaystyle\left(\frac{y}{5}\right)^2=1\)

Show Graph

The center is \((0,0)\), x-radius is \(2\), y-radius is \(5\)

\(\textbf{2)}\) \(\displaystyle\frac{(x-3)^2}{16}+\displaystyle\frac{(y+2)^2}{4}=1\)

Show Hint

This is the same as \(\displaystyle\left(\frac{x-3}{4}\right)^2+\displaystyle\left(\frac{y+2}{2}\right)^2=1\)

Show Graph

The center is \((3,-2)\), x-radius is \(4\) y-radius is \(2\)

\(\textbf{3)}\) Find the equation of an ellipse with vertices at \((3,9)\) and \((3,-1)\) and foci at \((3,8)\) and \((3,0)\).

Show Equation

The equation is \(\displaystyle\frac{(x-3)^2}{9}+\displaystyle\frac{(y-4)^2}{25}=1\)

For problems 4-7, use \(\displaystyle\frac{(x-3)^2}{16}+\displaystyle\frac{(y+2)^2}{4}=1\)

\(\textbf{4)}\) Find the center of \(\displaystyle\frac{(x-3)^2}{16}+\displaystyle\frac{(y+2)^2}{4}=1\).

Show Hint

\(\text{Hint: The center is } (h,k) \text{ in }\displaystyle\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\)

Show the Center

The center is \((3,-2)\)

\(\textbf{5)}\) Find the vertices of \(\displaystyle\frac{(x-3)^2}{16}+\displaystyle\frac{(y+2)^2}{4}=1\).

Show Hint

\(\text{Hint: The vertices are } (h+a,k) \text{ and } (h-a,k) \text{ in }\displaystyle\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\)

Show the Vertices

The vertices are \((-1,-2)\) and \((7,-2)\)

\(\textbf{6)}\) Find the Foci of \(\displaystyle\frac{(x-3)^2}{16}+\displaystyle\frac{(y+2)^2}{4}=1\).

Show the Foci

The foci are \((3-2\sqrt{3},-2)\) and \((3+2\sqrt{3},-2)\)

\(\textbf{7)}\) Find the length of the minor axis of \(\displaystyle\frac{(x-3)^2}{16}+\displaystyle\frac{(y+2)^2}{4}=1\).

Show Hint

\(\text{Hint: The length of the minor axis is } 2b \text{ in }\displaystyle\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\)

Show Answer

The length of minor axis is \(4\)

\(\textbf{8)}\) Use completing the square to rewrite the equation in standard form. \(9x^2+4y^2-8y=32\)

Show Equation

The equation is \(\displaystyle\frac{x^2}{4}+\displaystyle\frac{(y-1)^2}{9}=1\)

Show Work

\(\,\,\,\,\,\,9x^2+4y^2-8y=32\)

\(\,\,\,\,\,\,9\left(x\right)^2+4\left(y^2-2y+\text{___}\right)=32+4\left(\text{___}\right)\)

\(\,\,\,\,\,\,9\left(x\right)^2+4\left(y^2-2y+\underline{ 1 }\right)=32+4\left(\underline{ 1 }\right)\)

\(\,\,\,\,\,\,9\left(x\right)^2+4\left(y-1\right)^2=36\)

\(\,\,\,\,\,\,\displaystyle\frac{9\left(x\right)^2}{36}+\frac{4\left(y-1\right)^2}{36}=\frac{36}{36}\)

\(\,\,\,\,\,\,\displaystyle\frac{x^2}{4}+\frac{\left(y-1\right)^2}{9}=1\)

\(\textbf{9)}\) Write the equation of the following ellipse in standard form.

Show Equation

The equation is \(\displaystyle\frac{(x-2)^2}{16}+\displaystyle\frac{(y-3)^2}{4}=1\)

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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