Solve 11 x y= 1 1/9

Question

Solve the equation

Solve for x

Solve for y

x = \frac{10}{99 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 = \frac{2 55 sin ( 2 θ )}{33 ∣ sin ( 2 θ ) ∣} r = - \frac{2 55 sin ( 2 θ )}{33 ∣ sin ( 2 θ ) ∣}

Evaluate

11 x y = 1 \frac{1}{9}

Evaluate

More Steps

11 x y = \frac{10}{9}

Multiply both sides of the equation by LCD

11 x y \times 9 = \frac{10}{9} \times 9

Simplify the equation

99 x y = \frac{10}{9} \times 9

Simplify the equation

99 x y = 10

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

99 cos (θ) \times r sin (θ) \times r = 10

Factor the expression

99 cos (θ) sin (θ) \times r^{2} = 10

Simplify the expression

\frac{99}{2} sin (2 θ) \times r^{2} = 10

Divide the terms

r^{2} = \frac{20}{99 sin ( 2 θ )}

Evaluate the power

r = \pm \frac{20}{99 sin ( 2 θ )}

Simplify the expression

More Steps

r = \pm \frac{2 55 sin ( 2 θ )}{33 ∣ sin ( 2 θ ) ∣}

Solution

r = \frac{2 55 sin ( 2 θ )}{33 ∣ sin ( 2 θ ) ∣} r = - \frac{2 55 sin ( 2 θ )}{33 ∣ sin ( 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

11 x y = 1 \frac{1}{9}

Simplify the expression

11 x y = \frac{10}{9}

Take the derivative of both sides

\frac{d}{d x} (11 x y) = \frac{d}{d x} (\frac{10}{9})

Calculate the derivative

More Steps

11 y + 11 x \frac{d y}{d x} = \frac{d}{d x} (\frac{10}{9})

Calculate the derivative

11 y + 11 x \frac{d y}{d x} = 0

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

11 x \frac{d y}{d x} = 0 - 11 y

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

11 x \frac{d y}{d x} = - 11 y

Divide both sides

\frac{11 x \frac{d y}{d x}}{11 x} = \frac{- 11 y}{11 x}

Divide the numbers

\frac{d y}{d x} = \frac{- 11 y}{11 x}

Divide the numbers

More Steps

\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

More Steps

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

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Conic

\frac{( x ^{'} ) ^{2}}{\frac{20}{99}} - \frac{( y ^{'} ) ^{2}}{\frac{20}{99}} = 1

Evaluate

11 x y = 1 \frac{1}{9}

Move the expression to the left side

11 x y - 1 \frac{1}{9} = 0

Covert the mixed number to an improper fraction

More Steps

11 x y - \frac{10}{9} = 0

The coefficients A,B and C of the general equation are A= 0,B= 11 and C= 0

A = 0 B = 11 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}{11}

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 11 x y - \frac{10}{9} = 0

11 (x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2}) (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) - \frac{10}{9} = 0

Calculate

More Steps

\frac{11}{2} (x^{'})^{2} - \frac{11}{2} (y^{'})^{2} - \frac{10}{9} = 0

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

\frac{11}{2} (x^{'})^{2} - \frac{11}{2} (y^{'})^{2} = 0 - (- \frac{10}{9})

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

\frac{11}{2} (x^{'})^{2} - \frac{11}{2} (y^{'})^{2} = 0 + \frac{10}{9}

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

\frac{11}{2} (x^{'})^{2} - \frac{11}{2} (y^{'})^{2} = \frac{10}{9}

Multiply both sides of the equation by \frac{9}{10}

(\frac{11}{2} (x^{'})^{2} - \frac{11}{2} (y^{'})^{2}) \times \frac{9}{10} = \frac{10}{9} \times \frac{9}{10}

Multiply the terms

More Steps

\frac{99}{20} (x^{'})^{2} - \frac{99}{20} (y^{'})^{2} = \frac{10}{9} \times \frac{9}{10}

Multiply the terms

More Steps

\frac{99}{20} (x^{'})^{2} - \frac{99}{20} (y^{'})^{2} = 1

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

\frac{( x ^{'} ) ^{2}}{\frac{20}{99}} - \frac{99}{20} (y^{'})^{2} = 1

Solution

\frac{( x ^{'} ) ^{2}}{\frac{20}{99}} - \frac{( y ^{'} ) ^{2}}{\frac{20}{99}} = 1

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