Lesson

Notes

Area of Ellipses
\(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\)
\(\text{Area}=\pi a b\)

 

 

Practice Problems

Find the area of the given ellipses

\(\textbf{1)}\) \(\displaystyle\frac{(x-3)^2}{25}+\displaystyle\frac{(y+2)^2}{16}=1\)

Show Area

The area \(= 20\pi \text{ units}^2\)

Show Work

\(\,\,\,\,\,\text{Area of ellipse}=\pi\cdot a \cdot b\)
\(\,\,\,\,\,\text a= \sqrt{25}=5\)
\(\,\,\,\,\,\text b= \sqrt{16}=4\)
\(\,\,\,\,\,\text{Area of ellipse}=\pi\cdot (5) \cdot (4)\)
\(\,\,\,\,\,\text{Area of ellipse}=20\pi\)

 

\(\textbf{2)}\) \(\displaystyle\frac{x^2}{36}+\displaystyle\frac{y^2}{49}=1\)

Show Area

The area \(= 42\pi \text{ units}^2\)

Show Work

\(\,\,\,\,\,\text{Area of ellipse}=\pi\cdot a \cdot b\)
\(\,\,\,\,\,\text a= \sqrt{36}=6\)
\(\,\,\,\,\,\text b= \sqrt{49}=7\)
\(\,\,\,\,\,\text{Area of ellipse}=\pi\cdot (6) \cdot (7)\)
\(\,\,\,\,\,\text{Area of ellipse}=42\pi\)

 

\(\textbf{3)}\) \(\displaystyle\frac{(x+5)^2}{1}+\displaystyle\frac{(y-1)^2}{4}=1\)

Show Area

The area \(= 2\pi \text{ units}^2\)

Show Work

\(\,\,\,\,\,\text{Area of ellipse}=\pi\cdot a \cdot b\)
\(\,\,\,\,\,\text a= \sqrt{1}=1\)
\(\,\,\,\,\,\text b= \sqrt{4}=2\)
\(\,\,\,\,\,\text{Area of ellipse}=\pi\cdot (1) \cdot (2)\)
\(\,\,\,\,\,\text{Area of ellipse}=2\pi\)

 

\(\textbf{4)}\) \(4x^2+9y^2=36\)

Show Area

The area \(= 6\pi \text{ units}^2\)

 

\(\textbf{5)}\) \(16(x+1)^2+9y^2=144\)

Show Area

The area \(= 12\pi \text{ units}^2\)

 

\(\textbf{6)}\) \(\displaystyle\frac{(x+8)^2}{2}+\displaystyle\frac{(y-4)^2}{5}=1\)

Show Area

The area \(= \sqrt{10}\pi \text{ units}^2\)

Show Work

\(\,\,\,\,\,\text{Area of ellipse}=\pi\cdot a \cdot b\)
\(\,\,\,\,\,\text a= \sqrt{2}\)
\(\,\,\,\,\,\text b= \sqrt{5}\)
\(\,\,\,\,\,\text{Area of ellipse}=\pi\cdot \sqrt{2} \cdot \sqrt{5}\)
\(\,\,\,\,\,\text{Area of ellipse}=\sqrt{10}\pi\)

 

\(\textbf{7)}\) \(\displaystyle\frac{(x+8)^2}{6}+\displaystyle\frac{(y-4)^2}{10}=1\)

Show Area

The area \(= 2\sqrt{15}\pi \text{ units}^2\)

Show Work

\(\,\,\,\,\,\text{Area of ellipse}=\pi\cdot a \cdot b\)
\(\,\,\,\,\,\text a= \sqrt{6}\)
\(\,\,\,\,\,\text b= \sqrt{10}\)
\(\,\,\,\,\,\text{Area of ellipse}=\pi\cdot \sqrt{6} \cdot \sqrt{10}\)
\(\,\,\,\,\,\text{Area of ellipse}=\pi\cdot \sqrt{6 \cdot 10} \)
\(\,\,\,\,\,\text{Area of ellipse}=\pi\cdot \sqrt{2 \cdot 3 \cdot 2 \cdot 5} \)
\(\,\,\,\,\,\text{Area of ellipse}=2\sqrt{15}\pi\)

 

\(\textbf{8)}\)

Show Area

The area is \(8\pi\)

Show Work

\(\,\,\,\,\,\text{Area of ellipse}=\pi\cdot a \cdot b\)
\(\,\,\,\,\,\text a= 1/2 \text{ horizontal axis}= 1/2 \cdot 8=4\)
\(\,\,\,\,\,\text b= 1/2 \text{ vertical axis}= 1/2 \cdot 4=2\)
\(\,\,\,\,\,\text{Area of ellipse}=\pi\cdot (4) \cdot (2)\)
\(\,\,\,\,\,\text{Area of ellipse}=8\pi\)

 

See Related Pages\(\)

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

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

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

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

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

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

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

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

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

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