Solve x^29-y^216=1

Identify the conic

Find the standard equation of the hyperbola

Find the center of the hyperbola

Find the foci of the hyperbola

\frac{x ^{2}}{\frac{1}{9}} - \frac{y ^{2}}{\frac{1}{16}} = 1

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Solve the equation

Solve for x

Solve for y

x = \frac{1 + 16 y ^{2}}{3} x = - \frac{1 + 16 y ^{2}}{3}

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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{9 x}{16 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{144 y ^{2} - 81 x ^{2}}{256 y ^{3}}

Calculate

x^{2} 9 - y^{2} 16 = 1

Simplify the expression

9 x^{2} - 16 y^{2} = 1

Take the derivative of both sides

\frac{d}{d x} (9 x^{2} - 16 y^{2}) = \frac{d}{d x} (1)

Calculate the derivative

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18 x - 32 y \frac{d y}{d x} = \frac{d}{d x} (1)

Calculate the derivative

18 x - 32 y \frac{d y}{d x} = 0

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

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

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

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

Divide both sides

\frac{- 32 y \frac{d y}{d x}}{- 32 y} = \frac{- 18 x}{- 32 y}

Divide the numbers

\frac{d y}{d x} = \frac{- 18 x}{- 32 y}

Cancel out the common factor - 2

\frac{d y}{d x} = \frac{9 x}{16 y}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{9 x}{16 y})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{9 x}{16 y})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 9 x ) \times 16 y - 9 x \times \frac{d}{d x} ( 16 y )}{( 16 y ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{9 \times 16 y - 9 x \times \frac{d}{d x} ( 16 y )}{( 16 y ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{9 \times 16 y - 9 x \times 16 \frac{d y}{d x}}{( 16 y ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{144 y - 9 x \times 16 \frac{d y}{d x}}{( 16 y ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{144 y - 144 x \frac{d y}{d x}}{( 16 y ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{144 y - 144 x \frac{d y}{d x}}{256 y ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{9 y - 9 x \frac{d y}{d x}}{16 y ^{2}}

Use equation \frac{d y}{d x} = \frac{9 x}{16 y} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{9 y - 9 x \times \frac{9 x}{16 y}}{16 y ^{2}}

Solution

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\frac{d ^{2} y}{d x ^{2}} = \frac{144 y ^{2} - 81 x ^{2}}{256 y ^{3}}

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Rewrite the equation

r = \frac{25 cos ^{2} ( θ ) - 16}{∣ 25 cos ^{2} ( θ ) - 16 ∣} r = - \frac{25 cos ^{2} ( θ ) - 16}{∣ 25 cos ^{2} ( θ ) - 16 ∣}

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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