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\frac{x^2}{36}+\frac{y^2}{9} = 1

Algebra Calculus Trigonometry Matrix

Question

\frac{x^{2}}{36}+\frac{y^{2}}{9}=1

Identify the conic

\left(0,0\right)

Rewrite in standard form

\frac{x^{2}}{36}+\frac{y^{2}}{9}=1

Solution

\left(0,0\right)

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Find the center of the ellipse

Find the foci of the ellipse

Find the vertices of the ellipse

Find the eccentricity of the ellipse

Solve the equation

\begin{align}&x=2\sqrt{9-y^{2}}\\&x=-2\sqrt{9-y^{2}}\end{align}

Evaluate

\frac{x^{2}}{36}+\frac{y^{2}}{9}=1

Move the expression to the right-hand side and change its sign

\frac{x^{2}}{36}=1-\frac{y^{2}}{9}

Subtract the terms

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Evaluate

1-\frac{y^{2}}{9}

Reduce fractions to a common denominator

\frac{9}{9}-\frac{y^{2}}{9}

Write all numerators above the common denominator

\frac{9-y^{2}}{9}

\frac{x^{2}}{36}=\frac{9-y^{2}}{9}

\text{Multiply both sides of the equation by }36

\frac{x^{2}}{36}\times 36=\frac{9-y^{2}}{9}\times 36

Multiply the terms

x^{2}=\frac{\left(9-y^{2}\right)\times 36}{9}

Evaluate

x^{2}=36-4y^{2}

Take the root of both sides of the equation and remember to use both positive and negative roots

x=\pm \sqrt{36-4y^{2}}

Simplify the expression

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Evaluate

\sqrt{36-4y^{2}}

Factor the expression

\sqrt{4\left(9-y^{2}\right)}

The root of a product is equal to the product of the roots of each factor

\sqrt{4}\times \sqrt{9-y^{2}}

Evaluate the root

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Evaluate

\sqrt{4}

\text{Write the number in exponential form with the base of }2

\sqrt{2^{2}}

\text{Reduce the index of the radical and exponent with }2

2\sqrt{9-y^{2}}

x=\pm 2\sqrt{9-y^{2}}

Solution

\begin{align}&x=2\sqrt{9-y^{2}}\\&x=-2\sqrt{9-y^{2}}\end{align}

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\text{Solve for }x

\text{Solve for }y

Testing for symmetry

\textrm{Symmetry with respect to the origin}

Evaluate

\frac{x^{2}}{36}+\frac{y^{2}}{9}=1

\text{To test if the graph of }\frac{x^{2}}{36}+\frac{y^{2}}{9}=1\text{ is symmetry with respect to the origin,substitute -x for x and -y for y}

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

Evaluate

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Evaluate

\frac{\left(-x\right)^{2}}{36}+\frac{\left(-y\right)^{2}}{9}

Rewrite the expression

\frac{x^{2}}{36}+\frac{y^{2}}{9}

Reduce fractions to a common denominator

\frac{x^{2}}{36}+\frac{y^{2}\times 4}{9\times 4}

Multiply the numbers

\frac{x^{2}}{36}+\frac{y^{2}\times 4}{36}

Write all numerators above the common denominator

\frac{x^{2}+y^{2}\times 4}{36}

Use the commutative property to reorder the terms

\frac{x^{2}+4y^{2}}{36}

\frac{x^{2}+4y^{2}}{36}=1

Solution

\textrm{Symmetry with respect to the origin}

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Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Find the first derivative

\frac{dy}{dx}=-\frac{x}{4y}

Calculate

\frac{x^{2}}{36}+\frac{y^{2}}{9}=1

Take the derivative of both sides

\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)

Use differentiation rules

\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)

Rewrite the expression

\frac{\frac{d}{dx}\left(x^{2}\right)}{36}

\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}

\frac{2x}{36}

Calculate

\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)

Rewrite the expression

\frac{\frac{d}{dx}\left(y^{2}\right)}{9}

Evaluate the derivative

\frac{2y\frac{dy}{dx}}{9}

\frac{x}{18}+\frac{2y\frac{dy}{dx}}{9}

Calculate

\frac{x+4y\frac{dy}{dx}}{18}

\frac{x+4y\frac{dy}{dx}}{18}=\frac{d}{dx}\left(1\right)

Calculate the derivative

\frac{x+4y\frac{dy}{dx}}{18}=0

Simplify

x+4y\frac{dy}{dx}=0

Move the constant to the right side

4y\frac{dy}{dx}=0-x

Removing 0 doesn't change the value,so remove it from the expression

4y\frac{dy}{dx}=-x

Divide both sides

\frac{4y\frac{dy}{dx}}{4y}=\frac{-x}{4y}

Divide the numbers

\frac{dy}{dx}=\frac{-x}{4y}

Solution

\frac{dy}{dx}=-\frac{x}{4y}

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\text{Find the derivative with respect to }x

\text{Find the derivative with respect to }y

Find the second derivative

\frac{d^2y}{dx^2}=-\frac{4y^{2}+x^{2}}{16y^{3}}

Calculate

\frac{x^{2}}{36}+\frac{y^{2}}{9}=1

Take the derivative of both sides

\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)

Use differentiation rules

\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)

Rewrite the expression

\frac{\frac{d}{dx}\left(x^{2}\right)}{36}

\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}

\frac{2x}{36}

Calculate

\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)

Rewrite the expression

\frac{\frac{d}{dx}\left(y^{2}\right)}{9}

Evaluate the derivative

\frac{2y\frac{dy}{dx}}{9}

\frac{x}{18}+\frac{2y\frac{dy}{dx}}{9}

Calculate

\frac{x+4y\frac{dy}{dx}}{18}

\frac{x+4y\frac{dy}{dx}}{18}=\frac{d}{dx}\left(1\right)

Calculate the derivative

\frac{x+4y\frac{dy}{dx}}{18}=0

Simplify

x+4y\frac{dy}{dx}=0

Move the constant to the right side

4y\frac{dy}{dx}=0-x

Removing 0 doesn't change the value,so remove it from the expression

4y\frac{dy}{dx}=-x

Divide both sides

\frac{4y\frac{dy}{dx}}{4y}=\frac{-x}{4y}

Divide the numbers

\frac{dy}{dx}=\frac{-x}{4y}

\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}

\frac{dy}{dx}=-\frac{x}{4y}

Take the derivative of both sides

\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(-\frac{x}{4y}\right)

Calculate the derivative

\frac{d^2y}{dx^2}=\frac{d}{dx}\left(-\frac{x}{4y}\right)

Use differentiation rules

\frac{d^2y}{dx^2}=-\frac{\frac{d}{dx}\left(x\right)\times 4y-x\times \frac{d}{dx}\left(4y\right)}{\left(4y\right)^{2}}

\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}

\frac{d^2y}{dx^2}=-\frac{1\times 4y-x\times \frac{d}{dx}\left(4y\right)}{\left(4y\right)^{2}}

Calculate the derivative

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Evaluate

\frac{d}{dx}\left(4y\right)

Simplify

4\times \frac{d}{dx}\left(y\right)

Calculate

4\frac{dy}{dx}

\frac{d^2y}{dx^2}=-\frac{1\times 4y-x\times 4\frac{dy}{dx}}{\left(4y\right)^{2}}

Any expression multiplied by 1 remains the same

\frac{d^2y}{dx^2}=-\frac{4y-x\times 4\frac{dy}{dx}}{\left(4y\right)^{2}}

Use the commutative property to reorder the terms

\frac{d^2y}{dx^2}=-\frac{4y-4x\frac{dy}{dx}}{\left(4y\right)^{2}}

Calculate

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Evaluate

\left(4y\right)^{2}

Evaluate the power

4^{2}y^{2}

Evaluate the power

16y^{2}

\frac{d^2y}{dx^2}=-\frac{4y-4x\frac{dy}{dx}}{16y^{2}}

Calculate

\frac{d^2y}{dx^2}=-\frac{y-x\frac{dy}{dx}}{4y^{2}}

\text{Use equation }\frac{dy}{dx}=-\frac{x}{4y}\text{ to substitute}

\frac{d^2y}{dx^2}=-\frac{y-x\left(-\frac{x}{4y}\right)}{4y^{2}}

Solution

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Calculate

-\frac{y-x\left(-\frac{x}{4y}\right)}{4y^{2}}

Multiply the terms

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Evaluate

x\left(-\frac{x}{4y}\right)

Multiplying or dividing an odd number of negative terms equals a negative

-x\times \frac{x}{4y}

Multiply the terms

-\frac{x\times x}{4y}

Multiply the terms

-\frac{x^{2}}{4y}

-\frac{y-\left(-\frac{x^{2}}{4y}\right)}{4y^{2}}

Subtract the terms

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Simplify

y-\left(-\frac{x^{2}}{4y}\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

y+\frac{x^{2}}{4y}

Reduce fractions to a common denominator

\frac{y\times 4y}{4y}+\frac{x^{2}}{4y}

Write all numerators above the common denominator

\frac{y\times 4y+x^{2}}{4y}

Multiply the terms

\frac{4y^{2}+x^{2}}{4y}

-\frac{\frac{4y^{2}+x^{2}}{4y}}{4y^{2}}

Divide the terms

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Evaluate

\frac{\frac{4y^{2}+x^{2}}{4y}}{4y^{2}}

Multiply by the reciprocal

\frac{4y^{2}+x^{2}}{4y}\times \frac{1}{4y^{2}}

Multiply the terms

\frac{4y^{2}+x^{2}}{4y\times 4y^{2}}

Multiply the terms

\frac{4y^{2}+x^{2}}{16y^{3}}

-\frac{4y^{2}+x^{2}}{16y^{3}}

\frac{d^2y}{dx^2}=-\frac{4y^{2}+x^{2}}{16y^{3}}

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\text{Find the second derivative with respect to }x

\text{Find the second derivative with respect to }y

Rewrite the equation

\begin{align}&r=\frac{6\sqrt{1+3\sin^{2}\left(\theta \right)}}{1+3\sin^{2}\left(\theta \right)}\\&r=-\frac{6\sqrt{1+3\sin^{2}\left(\theta \right)}}{1+3\sin^{2}\left(\theta \right)}\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

Simplify the equation

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Evaluate

\left(\frac{x^{2}}{36}+\frac{y^{2}}{9}\right)\times 36

Apply the distributive property

\frac{x^{2}}{36}\times 36+\frac{y^{2}}{9}\times 36

Simplify

x^{2}+y^{2}\times 4

Use the commutative property to reorder the terms

x^{2}+4y^{2}

x^{2}+4y^{2}=1\times 36

Any expression multiplied by 1 remains the same

x^{2}+4y^{2}=36

\text{To convert the equation to polar coordinates,substitute }x\text{ for }r\cos\left(\theta \right)\text{ and }y\text{ for }r\sin\left(\theta \right)

\left(\cos\left(\theta \right)\times r\right)^{2}+4\left(\sin\left(\theta \right)\times r\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

Divide the terms

r^{2}=\frac{36}{-3\cos^{2}\left(\theta \right)+4}

Simplify the expression

r^{2}=\frac{36}{1+3\sin^{2}\left(\theta \right)}

Evaluate the power

r=\pm \sqrt{\frac{36}{1+3\sin^{2}\left(\theta \right)}}

Simplify the expression

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Evaluate

\sqrt{\frac{36}{1+3\sin^{2}\left(\theta \right)}}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{\sqrt{36}}{\sqrt{1+3\sin^{2}\left(\theta \right)}}

Simplify the radical expression

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Evaluate

\sqrt{36}

\text{Write the number in exponential form with the base of }6

\sqrt{6^{2}}

\text{Reduce the index of the radical and exponent with }2

\frac{6}{\sqrt{1+3\sin^{2}\left(\theta \right)}}

Multiply by the Conjugate

\frac{6\sqrt{1+3\sin^{2}\left(\theta \right)}}{\sqrt{1+3\sin^{2}\left(\theta \right)}\times \sqrt{1+3\sin^{2}\left(\theta \right)}}

Calculate

\frac{6\sqrt{1+3\sin^{2}\left(\theta \right)}}{1+3\sin^{2}\left(\theta \right)}

r=\pm \frac{6\sqrt{1+3\sin^{2}\left(\theta \right)}}{1+3\sin^{2}\left(\theta \right)}

Solution

\begin{align}&r=\frac{6\sqrt{1+3\sin^{2}\left(\theta \right)}}{1+3\sin^{2}\left(\theta \right)}\\&r=-\frac{6\sqrt{1+3\sin^{2}\left(\theta \right)}}{1+3\sin^{2}\left(\theta \right)}\end{align}

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