\frac{x^2}{36}+\frac{y^2}{9} = 1
Question
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Identify the conic
Write the equation in standard form
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Solution
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Find the center of the ellipse
\left(0,0\right)
Evaluate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Solution
\left(0,0\right)
Find the focus
F_{1} = \left(-3\sqrt{3},0\right), F_{2} = \left(3\sqrt{3},0\right)
Evaluate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Solution
F_{1} = \left(-3\sqrt{3},0\right), F_{2} = \left(3\sqrt{3},0\right)
Find the vertices
V_{1} = \left(0,3\right), V_{2} = \left(0,-3\right), V_{3} = \left(6,0\right), V_{4} = \left(-6,0\right)
Evaluate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Solution
V_{1} = \left(0,3\right), V_{2} = \left(0,-3\right), V_{3} = \left(6,0\right), V_{4} = \left(-6,0\right)
Find the eccentricity
e = \frac{\sqrt{3}}{2}
Evaluate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Solution
e = \frac{\sqrt{3}}{2}
Choose Method
Write the equation in standard form
Find the center of the ellipse
Find the focus
Find the vertices
Find the eccentricity
Solve the equation
\text{Solve for }x
\begin{align}&x=2\sqrt{-y^{2}+9}\\&x=-2\sqrt{-y^{2}+9}\end{align}
Evaluate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Move the expression to the left side
\frac{x^{2}}{36}+\frac{y^{2}}{9}-1=0
\text{Substitute a=}\frac{1}{36}\text{,b=}0\text{ and c=}\frac{y^{2}}{9}-1\text{ into the quadratic formula }x\text{=}\frac{-b\pm\sqrt{b^2-4ac}}{2a}
x=\frac{0\pm \sqrt{0^{2}-4\times \frac{1}{36}\left(\frac{y^{2}}{9}-1\right)}}{2\times \frac{1}{36}}
Simplify the expression
x=\frac{0\pm \sqrt{0^{2}-4\times \frac{1}{36}\left(\frac{y^{2}}{9}-1\right)}}{\frac{1}{18}}
Simplify the expression
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Evaluate
0^{2}-4\times \frac{1}{36}\left(\frac{y^{2}}{9}-1\right)
Calculate
0-4\times \frac{1}{36}\left(\frac{y^{2}}{9}-1\right)
Multiply the terms
0-\left(\frac{y^{2}}{81}-\frac{1}{9}\right)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
0-\frac{y^{2}}{81}+\frac{1}{9}
-\frac{y^{2}}{81}+\frac{1}{9}
x=\frac{0\pm \sqrt{-\frac{y^{2}}{81}+\frac{1}{9}}}{\frac{1}{18}}
\text{Separate the equation into }2\text{ possible cases}
\begin{align}&x=\frac{0+\sqrt{-\frac{y^{2}}{81}+\frac{1}{9}}}{\frac{1}{18}}\\&x=\frac{0-\sqrt{-\frac{y^{2}}{81}+\frac{1}{9}}}{\frac{1}{18}}\end{align}
Solve the equation
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Evaluate
x=\frac{0+\sqrt{-\frac{y^{2}}{81}+\frac{1}{9}}}{\frac{1}{18}}
x=\frac{\sqrt{-\frac{y^{2}}{81}+\frac{1}{9}}}{\frac{1}{18}}
Calculate
x=18\sqrt{-\frac{y^{2}}{81}+\frac{1}{9}}
\begin{align}&x=18\sqrt{-\frac{y^{2}}{81}+\frac{1}{9}}\\&x=\frac{0-\sqrt{-\frac{y^{2}}{81}+\frac{1}{9}}}{\frac{1}{18}}\end{align}
Calculate
\begin{align}&x=18\sqrt{-\frac{y^{2}}{81}+\frac{1}{9}}\\&x=-18\sqrt{-\frac{y^{2}}{81}+\frac{1}{9}}\end{align}
Solution
\begin{align}&x=2\sqrt{-y^{2}+9}\\&x=-2\sqrt{-y^{2}+9}\end{align}
\text{Solve for }y
\begin{align}&y=\frac{\sqrt{-x^{2}+36}}{2}\\&y=-\frac{\sqrt{-x^{2}+36}}{2}\end{align}
Evaluate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Move the expression to the left side
\frac{x^{2}}{36}+\frac{y^{2}}{9}-1=0
\frac{x^{2}}{36}-1+\frac{y^{2}}{9}=0
Rewrite in standard form
\frac{y^{2}}{9}+\frac{x^{2}}{36}-1=0
\text{Substitute a=}\frac{1}{9}\text{,b=}0\text{ and c=}\frac{x^{2}}{36}-1\text{ into the quadratic formula }y\text{=}\frac{-b\pm\sqrt{b^2-4ac}}{2a}
y=\frac{0\pm \sqrt{0^{2}-4\times \frac{1}{9}\left(\frac{x^{2}}{36}-1\right)}}{2\times \frac{1}{9}}
Simplify the expression
y=\frac{0\pm \sqrt{0^{2}-4\times \frac{1}{9}\left(\frac{x^{2}}{36}-1\right)}}{\frac{2}{9}}
Simplify the expression
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Evaluate
0^{2}-4\times \frac{1}{9}\left(\frac{x^{2}}{36}-1\right)
Calculate
0-4\times \frac{1}{9}\left(\frac{x^{2}}{36}-1\right)
Multiply the terms
0-\left(\frac{x^{2}}{81}-\frac{4}{9}\right)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
0-\frac{x^{2}}{81}+\frac{4}{9}
-\frac{x^{2}}{81}+\frac{4}{9}
y=\frac{0\pm \sqrt{-\frac{x^{2}}{81}+\frac{4}{9}}}{\frac{2}{9}}
\text{Separate the equation into }2\text{ possible cases}
\begin{align}&y=\frac{0+\sqrt{-\frac{x^{2}}{81}+\frac{4}{9}}}{\frac{2}{9}}\\&y=\frac{0-\sqrt{-\frac{x^{2}}{81}+\frac{4}{9}}}{\frac{2}{9}}\end{align}
Solve the equation
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Evaluate
y=\frac{0+\sqrt{-\frac{x^{2}}{81}+\frac{4}{9}}}{\frac{2}{9}}
y=\frac{\sqrt{-\frac{x^{2}}{81}+\frac{4}{9}}}{\frac{2}{9}}
Calculate
y=\frac{9\sqrt{-\frac{x^{2}}{81}+\frac{4}{9}}}{2}
\begin{align}&y=\frac{9\sqrt{-\frac{x^{2}}{81}+\frac{4}{9}}}{2}\\&y=\frac{0-\sqrt{-\frac{x^{2}}{81}+\frac{4}{9}}}{\frac{2}{9}}\end{align}
Calculate
\begin{align}&y=\frac{9\sqrt{-\frac{x^{2}}{81}+\frac{4}{9}}}{2}\\&y=-\frac{9\sqrt{-\frac{x^{2}}{81}+\frac{4}{9}}}{2}\end{align}
Solution
\begin{align}&y=\frac{\sqrt{-x^{2}+36}}{2}\\&y=-\frac{\sqrt{-x^{2}+36}}{2}\end{align}
Choose Method
\text{Solve for }x
\text{Solve for }y
Rewrite the equation
Rewrite in polar form
\begin{align}&r=\frac{6}{\sqrt{-3\cos^{2}\left(\theta \right)+4}}\\&r=-\frac{6}{\sqrt{-3\cos^{2}\left(\theta \right)+4}}\end{align}
Evaluate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Multiply both sides of the equation by LCD
\left(\frac{x^{2}}{36}+\frac{y^{2}}{9}\right)\times 36=1\times 36
Evaluate
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Evaluate
\left(\frac{x^{2}}{36}+\frac{y^{2}}{9}\right)\times 36
Evaluate
\frac{x^{2}}{36}\times 36+\frac{y^{2}}{9}\times 36
Evaluate
x^{2}+y^{2}\times 4
Evaluate
x^{2}+4y^{2}
x^{2}+4y^{2}=1\times 36
Evaluate
x^{2}+4y^{2}=36
\text{To convert the equation to polar coordinates,substitute }x\text{ for }r\times \cos\left(\theta \right)\text{ and }y\text{ for }r\times \sin\left(\theta \right)
\left(r\cos\left(\theta \right)\right)^{2}+4\left(r\sin\left(\theta \right)\right)^{2}=36
Factor the expression
\left(\cos^{2}\left(\theta \right)+4\sin^{2}\left(\theta \right)\right)r^{2}=36
Simplify the expression
\left(-3\cos^{2}\left(\theta \right)+4\right)r^{2}=36
Factor the expression
r^{2}\times \left(-3\cos^{2}\left(\theta \right)+4\right)=36
Divide the terms
r^{2}=-\frac{36}{3\cos^{2}\left(\theta \right)-4}
Evaluate the power
r=\pm \sqrt{-\frac{36}{3\cos^{2}\left(\theta \right)-4}}
Simplify the expression
r=\pm \frac{6}{\sqrt{-3\cos^{2}\left(\theta \right)+4}}
Solution
\begin{align}&r=\frac{6}{\sqrt{-3\cos^{2}\left(\theta \right)+4}}\\&r=-\frac{6}{\sqrt{-3\cos^{2}\left(\theta \right)+4}}\end{align}
Find the derivative
\text{Find the derivative with respect to }x
\frac{dy}{dx}=-\frac{x}{4y}
Evaluate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Take the derivative
\frac{d}{dx}\left(\frac{x^{2}}{36}+\frac{y^{2}}{9}\right)=\frac{d}{dx}\left(1\right)
Calculate the derivative
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Evaluate
\frac{d}{dx}\left(\frac{x^{2}}{36}+\frac{y^{2}}{9}\right)
Evaluate the derivative
\frac{d}{dx}\left(\frac{x^{2}}{36}\right)+\frac{d}{dx}\left(\frac{y^{2}}{9}\right)
Evaluate the derivative
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Evaluate
\frac{d}{dx}\left(\frac{x^{2}}{36}\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
\frac{1}{36}\times \frac{d}{dx}\left(x^{2}\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
\frac{1}{36}\times 2x
Multiply the terms
\frac{x}{18}
\frac{x}{18}+\frac{d}{dx}\left(\frac{y^{2}}{9}\right)
Evaluate the derivative
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Evaluate
\frac{d}{dx}\left(\frac{y^{2}}{9}\right)
Use differentiation rules
\frac{d}{dy}\left(\frac{y^{2}}{9}\right)\times \frac{dy}{dx}
Evaluate the derivative
\frac{2y}{9}\times \frac{dy}{dx}
Multiply the terms
\frac{2y\frac{dy}{dx}}{9}
\frac{x}{18}+\frac{2y\frac{dy}{dx}}{9}
\frac{x}{18}+\frac{2y\frac{dy}{dx}}{9}=\frac{d}{dx}\left(1\right)
Calculate the derivative
\frac{x}{18}+\frac{2y\frac{dy}{dx}}{9}=0
Rewrite the expression
\frac{2y}{9}\times \frac{dy}{dx}=0-\frac{x}{18}
Simplify
\frac{2y}{9}\times \frac{dy}{dx}=-\frac{x}{18}
Divide both sides
\frac{dy}{dx}=\frac{-\frac{x}{18}}{\frac{2y}{9}}
Solution
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Evaluate
\frac{-\frac{x}{18}}{\frac{2y}{9}}
Multiply by the reciprocal
-\frac{x}{18}\times \frac{9}{2y}
Multiply the terms
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Evaluate
-\frac{x}{18}\times \frac{9}{2y}
Factor the expression
\frac{-x}{18}\times \frac{9}{2y}
Reduce the fraction
\frac{-x}{2}\times \frac{1}{2y}
Multiply the terms
\frac{-x}{2\times 2y}
Multiply the terms
\frac{-x}{4y}
Calculate
-\frac{x}{4y}
-\frac{x}{4y}
\frac{dy}{dx}=-\frac{x}{4y}
\text{Find the derivative with respect to }y
\frac{dx}{dy}=-\frac{4y}{x}
Evaluate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Take the derivative
\frac{d}{dy}\left(\frac{x^{2}}{36}+\frac{y^{2}}{9}\right)=\frac{d}{dy}\left(1\right)
Calculate the derivative
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Evaluate
\frac{d}{dy}\left(\frac{x^{2}}{36}+\frac{y^{2}}{9}\right)
Evaluate the derivative
\frac{d}{dy}\left(\frac{x^{2}}{36}\right)+\frac{d}{dy}\left(\frac{y^{2}}{9}\right)
Evaluate the derivative
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Evaluate
\frac{d}{dy}\left(\frac{x^{2}}{36}\right)
Use differentiation rules
\frac{d}{dx}\left(\frac{x^{2}}{36}\right)\times \frac{dx}{dy}
Evaluate the derivative
\frac{x}{18}\times \frac{dx}{dy}
Multiply the terms
\frac{x\frac{dx}{dy}}{18}
\frac{x\frac{dx}{dy}}{18}+\frac{d}{dy}\left(\frac{y^{2}}{9}\right)
Evaluate the derivative
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Evaluate
\frac{d}{dy}\left(\frac{y^{2}}{9}\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
\frac{1}{9}\times \frac{d}{dy}\left(y^{2}\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
\frac{1}{9}\times 2y
Multiply the terms
\frac{2y}{9}
\frac{x\frac{dx}{dy}}{18}+\frac{2y}{9}
\frac{x\frac{dx}{dy}}{18}+\frac{2y}{9}=\frac{d}{dy}\left(1\right)
Calculate the derivative
\frac{x\frac{dx}{dy}}{18}+\frac{2y}{9}=0
Rewrite the expression
\frac{x}{18}\times \frac{dx}{dy}=0-\frac{2y}{9}
Simplify
\frac{x}{18}\times \frac{dx}{dy}=-\frac{2y}{9}
Divide both sides
\frac{dx}{dy}=\frac{-\frac{2y}{9}}{\frac{x}{18}}
Solution
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Evaluate
\frac{-\frac{2y}{9}}{\frac{x}{18}}
Multiply by the reciprocal
-\frac{2y}{9}\times \frac{18}{x}
Multiply the terms
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Evaluate
-\frac{2y}{9}\times \frac{18}{x}
Factor the expression
\frac{-2y}{9}\times \frac{18}{x}
Reduce the fraction
-2y\times \frac{2}{x}
Multiply the terms
\frac{-2y\times 2}{x}
Multiply the terms
\frac{-4y}{x}
Calculate
-\frac{4y}{x}
-\frac{4y}{x}
\frac{dx}{dy}=-\frac{4y}{x}
Choose Method
\text{Find the derivative with respect to }x
\text{Find the derivative with respect to }y
Graph