Solve x^24 + y^29 = 1

Identify the conic

Find the standard equation of the ellipse

Find the center of the ellipse

Find the foci of the ellipse

\frac{x ^{2}}{\frac{1}{4}} + \frac{y ^{2}}{\frac{1}{9}} = 1

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Solve the equation

Solve for x

Solve for y

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

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Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Symmetry with respect to the origin

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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{4 x}{9 y}

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Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = - \frac{36 y ^{2} + 16 x ^{2}}{81 y ^{3}}

Calculate

x^{2} 4 + y^{2} 9 = 1

Simplify the expression

4 x^{2} + 9 y^{2} = 1

Take the derivative of both sides

\frac{d}{d x} (4 x^{2} + 9 y^{2}) = \frac{d}{d x} (1)

Calculate the derivative

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8 x + 18 y \frac{d y}{d x} = \frac{d}{d x} (1)

Calculate the derivative

8 x + 18 y \frac{d y}{d x} = 0

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

18 y \frac{d y}{d x} = 0 - 8 x

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

18 y \frac{d y}{d x} = - 8 x

Divide both sides

\frac{18 y \frac{d y}{d x}}{18 y} = \frac{- 8 x}{18 y}

Divide the numbers

\frac{d y}{d x} = \frac{- 8 x}{18 y}

Divide the numbers

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\frac{d y}{d x} = - \frac{4 x}{9 y}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{4 x}{9 y})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{4 x}{9 y})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( 4 x ) \times 9 y - 4 x \times \frac{d}{d x} ( 9 y )}{( 9 y ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = - \frac{4 \times 9 y - 4 x \times \frac{d}{d x} ( 9 y )}{( 9 y ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = - \frac{4 \times 9 y - 4 x \times 9 \frac{d y}{d x}}{( 9 y ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{36 y - 4 x \times 9 \frac{d y}{d x}}{( 9 y ) ^{2}}

Calculate

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

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{36 y - 36 x \frac{d y}{d x}}{81 y ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y - 4 x \frac{d y}{d x}}{9 y ^{2}}

Use equation \frac{d y}{d x} = - \frac{4 x}{9 y} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y - 4 x ( - \frac{4 x}{9 y} )}{9 y ^{2}}

Solution

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\frac{d ^{2} y}{d x ^{2}} = - \frac{36 y ^{2} + 16 x ^{2}}{81 y ^{3}}

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Rewrite the equation

r = \frac{4 + 5 sin ^{2} ( θ )}{4 + 5 sin ^{2} ( θ )} r = - \frac{4 + 5 sin ^{2} ( θ )}{4 + 5 sin ^{2} ( θ )}

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Identify the conic

Solve the equation

Testing for symmetry

Find the first derivative

Find the second derivative

Rewrite the equation

Graph