Solve x^2 - 2y^2 = 1

Identify the conic

Find the standard equation of the hyperbola

Find the center of the hyperbola

Find the foci of the hyperbola

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

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Solve for x

Solve for y

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

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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}{2 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{2 y ^{2} - x ^{2}}{4 y ^{3}}

Calculate

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

Take the derivative of both sides

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

Calculate the derivative

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

Calculate the derivative

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

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

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

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

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

Divide both sides

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

Divide the numbers

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

Cancel out the common factor - 2

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Any expression multiplied by 1 remains the same

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

Use the commutative property to reorder the terms

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

Calculate

More Steps

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

Calculate

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

Use equation \frac{d y}{d x} = \frac{x}{2 y} to substitute

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

Solution

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

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r = \frac{3 cos ^{2} ( θ ) - 2}{∣ 3 cos ^{2} ( θ ) - 2 ∣} r = - \frac{3 cos ^{2} ( θ ) - 2}{∣ 3 cos ^{2} ( θ ) - 2 ∣}

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