Solve x^2 - 4y^2-4 =0

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}}{4} - y^{2} = 1

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Solve for x

Solve for y

x = 2 y^{2} + 1 x = - 2 y^{2} + 1

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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 derivative with respect to x

Find the derivative with respect to y

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

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Find the second derivative with respect to x

Find the second derivative with respect to y

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

Calculate

x^{2} - 4 y^{2} - 4 = 0

Take the derivative of both sides

\frac{d}{d x} (x^{2} - 4 y^{2} - 4) = \frac{d}{d x} (0)

Calculate the derivative

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2 x - 8 y \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

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

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

- 8 y \frac{d y}{d x} = 0 - 2 x

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

- 8 y \frac{d y}{d x} = - 2 x

Divide both sides

\frac{- 8 y \frac{d y}{d x}}{- 8 y} = \frac{- 2 x}{- 8 y}

Divide the numbers

\frac{d y}{d x} = \frac{- 2 x}{- 8 y}

Cancel out the common factor - 2

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

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

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

Calculate the derivative

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

Any expression multiplied by 1 remains the same

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

Use the commutative property to reorder the terms

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

Calculate

More Steps

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

Calculate

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

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

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

Solution

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

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r = \frac{2 5 cos ^{2} ( θ ) - 4}{∣ 5 cos ^{2} ( θ ) - 4 ∣} r = - \frac{2 5 cos ^{2} ( θ ) - 4}{∣ 5 cos ^{2} ( θ ) - 4 ∣}

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