Solve y^2-4x^27=0

Solve the equation

Solve for x

Solve for y

x = \frac{7 \times ∣ y ∣}{14} x = - \frac{7 \times ∣ y ∣}{14}

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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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Rewrite the equation

r = 0 θ = ⎩ ⎨ ⎧ - arcsin (\frac{2 203}{29}) + kπ arcsin (\frac{2 203}{29}) + kπ, k \in Z

Evaluate

y^{2} - 4 x^{2} \times 7 = 0

Evaluate

y^{2} - 28 x^{2} = 0

To convert the equation to polar coordinates,substitute x for r cos (θ) and y for r sin (θ)

(sin (θ) \times r)^{2} - 28 (cos (θ) \times r)^{2} = 0

Factor the expression

(sin^{2} (θ) - 28 cos^{2} (θ)) r^{2} = 0

Simplify the expression

(29 sin^{2} (θ) - 28) r^{2} = 0

Separate into possible cases

r^{2} = 0 29 sin^{2} (θ) - 28 = 0

Evaluate

r = 0 29 sin^{2} (θ) - 28 = 0

Solution

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Evaluate

29 sin^{2} (θ) - 28 = 0

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

29 sin^{2} (θ) = 0 + 28

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

29 sin^{2} (θ) = 28

Divide both sides

\frac{29 sin ^{2} ( θ )}{29} = \frac{28}{29}

Divide the numbers

sin^{2} (θ) = \frac{28}{29}

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

sin (θ) = \pm \frac{28}{29}

Simplify the expression

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sin (θ) = \pm \frac{2 203}{29}

Separate the equation into 2 possible cases

sin (θ) = \frac{2 203}{29} sin (θ) = - \frac{2 203}{29}

Calculate

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θ = ⎩ ⎨ ⎧ arcsin (\frac{2 203}{29}) + 2 kπ - arcsin (\frac{2 203}{29}) + π + 2 kπ, k \in Z sin (θ) = - \frac{2 203}{29}

Calculate

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θ = ⎩ ⎨ ⎧ arcsin (\frac{2 203}{29}) + 2 kπ - arcsin (\frac{2 203}{29}) + π + 2 kπ, k \in Z θ = ⎩ ⎨ ⎧ - arcsin (\frac{2 203}{29}) + 2 kπ arcsin (\frac{2 203}{29}) + π + 2 kπ, k \in Z

Find the union

θ = ⎩ ⎨ ⎧ - arcsin (\frac{2 203}{29}) + kπ arcsin (\frac{2 203}{29}) + kπ, k \in Z

r = 0 θ = ⎩ ⎨ ⎧ - arcsin (\frac{2 203}{29}) + kπ arcsin (\frac{2 203}{29}) + kπ, k \in Z

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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{28 x}{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{28 y ^{2} - 784 x ^{2}}{y ^{3}}

Calculate

y^{2} - 4 x^{2} 7 = 0

Simplify the expression

y^{2} - 28 x^{2} = 0

Take the derivative of both sides

\frac{d}{d x} (y^{2} - 28 x^{2}) = \frac{d}{d x} (0)

Calculate the derivative

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2 y \frac{d y}{d x} - 56 x = \frac{d}{d x} (0)

Calculate the derivative

2 y \frac{d y}{d x} - 56 x = 0

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

2 y \frac{d y}{d x} = 0 + 56 x

Add the terms

2 y \frac{d y}{d x} = 56 x

Divide both sides

\frac{2 y \frac{d y}{d x}}{2 y} = \frac{56 x}{2 y}

Divide the numbers

\frac{d y}{d x} = \frac{56 x}{2 y}

Cancel out the common factor 2

\frac{d y}{d x} = \frac{28 x}{y}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{28 x}{y})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{28 x}{y})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 28 x ) \times y - 28 x \times \frac{d}{d x} ( y )}{y ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{28 y - 28 x \times \frac{d}{d x} ( y )}{y ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{28 y - 28 x \frac{d y}{d x}}{y ^{2}}

Use equation \frac{d y}{d x} = \frac{28 x}{y} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{28 y - 28 x \times \frac{28 x}{y}}{y ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{28 y ^{2} - 784 x ^{2}}{y ^{3}}

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