Solve 4x 8y= −48

Question

Solve the equation

Solve for x

Solve for y

x = - \frac{3}{2 y}

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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 = - 3 csc (2 θ) r = - - 3 csc (2 θ)

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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{y}{x}

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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{2 y}{x ^{2}}

Calculate

4 x 8 y = - 48

Simplify the expression

32 x y = - 48

Take the derivative of both sides

\frac{d}{d x} (32 x y) = \frac{d}{d x} (- 48)

Calculate the derivative

More Steps

32 y + 32 x \frac{d y}{d x} = \frac{d}{d x} (- 48)

Calculate the derivative

32 y + 32 x \frac{d y}{d x} = 0

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

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

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

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

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

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

Use the commutative property to reorder the terms

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

Any expression multiplied by 1 remains the same

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

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

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

Solution

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

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Conic

\frac{( y ^{'} ) ^{2}}{3} - \frac{( x ^{'} ) ^{2}}{3} = 1

Evaluate

4 x \times 8 y = - 48

Move the expression to the left side

4 x \times 8 y - (- 48) = 0

Calculate

More Steps

32 x y + 48 = 0

The coefficients A,B and C of the general equation are A= 0,B= 32 and C= 0

A = 0 B = 32 C = 0

To find the angle of rotation θ,substitute the values of A,B and C into the formula cot (2 θ) = \frac{A - C}{B}

cot (2 θ) = \frac{0 - 0}{32}

Calculate

cot (2 θ) = 0

Using the unit circle,find the smallest positive angle for which the cotangent is 0

2 θ = \frac{π}{2}

Calculate

θ = \frac{π}{4}

To rotate the axes,use the equation of rotation and substitute \frac{π}{4} for θ

x = x^{'} cos (\frac{π}{4}) - y^{'} sin (\frac{π}{4}) y = x^{'} sin (\frac{π}{4}) + y^{'} cos (\frac{π}{4})

Calculate

x = x^{'} \times \frac{2}{2} - y^{'} sin (\frac{π}{4}) y = x^{'} sin (\frac{π}{4}) + y^{'} cos (\frac{π}{4})

Calculate

x = x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2} y = x^{'} sin (\frac{π}{4}) + y^{'} cos (\frac{π}{4})

Calculate

x = x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2} y = x^{'} \times \frac{2}{2} + y^{'} cos (\frac{π}{4})

Calculate

x = x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2} y = x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}

Substitute x and y into the original equation 32 x y + 48 = 0

32 (x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2}) (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) + 48 = 0

Calculate

More Steps

16 (x^{'})^{2} - 16 (y^{'})^{2} + 48 = 0

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

16 (x^{'})^{2} - 16 (y^{'})^{2} = 0 - 48

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

16 (x^{'})^{2} - 16 (y^{'})^{2} = - 48

Multiply both sides of the equation by - \frac{1}{48}

(16 (x^{'})^{2} - 16 (y^{'})^{2}) (- \frac{1}{48}) = - 48 (- \frac{1}{48})

Multiply the terms

More Steps

- \frac{1}{3} (x^{'})^{2} + \frac{1}{3} (y^{'})^{2} = - 48 (- \frac{1}{48})

Multiply the terms

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- \frac{1}{3} (x^{'})^{2} + \frac{1}{3} (y^{'})^{2} = 1

Use a = \frac{1}{\frac{1}{a}} to transform the expression

- \frac{( x ^{'} ) ^{2}}{3} + \frac{1}{3} (y^{'})^{2} = 1

Use a = \frac{1}{\frac{1}{a}} to transform the expression

- \frac{( x ^{'} ) ^{2}}{3} + \frac{( y ^{'} ) ^{2}}{3} = 1

Solution

\frac{( y ^{'} ) ^{2}}{3} - \frac{( x ^{'} ) ^{2}}{3} = 1

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