Solve 2000x 1000y=3200

Question

Solve the equation

Solve for x

Solve for y

x = \frac{1}{625 y}

Show Solution

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

Show Solution

Rewrite the equation

r = \frac{2 sin ( 2 θ )}{25 ∣ sin ( 2 θ ) ∣} r = - \frac{2 sin ( 2 θ )}{25 ∣ sin ( 2 θ ) ∣}

Show Solution

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}

Show Solution

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

2000 x 1000 y = 3200

Simplify the expression

2000000 x y = 3200

Take the derivative of both sides

\frac{d}{d x} (2000000 x y) = \frac{d}{d x} (3200)

Calculate the derivative

More Steps

2000000 y + 2000000 x \frac{d y}{d x} = \frac{d}{d x} (3200)

Calculate the derivative

2000000 y + 2000000 x \frac{d y}{d x} = 0

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

2000000 x \frac{d y}{d x} = 0 - 2000000 y

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

2000000 x \frac{d y}{d x} = - 2000000 y

Divide both sides

\frac{2000000 x \frac{d y}{d x}}{2000000 x} = \frac{- 2000000 y}{2000000 x}

Divide the numbers

\frac{d y}{d x} = \frac{- 2000000 y}{2000000 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}}

Show Solution

Conic

\frac{( x ^{'} ) ^{2}}{\frac{2}{625}} - \frac{( y ^{'} ) ^{2}}{\frac{2}{625}} = 1

Evaluate

2000 x \times 1000 y = 3200

Move the expression to the left side

2000 x \times 1000 y - 3200 = 0

Calculate

2000000 x y - 3200 = 0

The coefficients A,B and C of the general equation are A= 0,B= 2000000 and C= 0

A = 0 B = 2000000 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}{2000000}

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 2000000 x y - 3200 = 0

2000000 (x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2}) (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) - 3200 = 0

Calculate

More Steps

1000000 (x^{'})^{2} - 1000000 (y^{'})^{2} - 3200 = 0

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

1000000 (x^{'})^{2} - 1000000 (y^{'})^{2} = 0 - (- 3200)

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

1000000 (x^{'})^{2} - 1000000 (y^{'})^{2} = 0 + 3200

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

1000000 (x^{'})^{2} - 1000000 (y^{'})^{2} = 3200

Multiply both sides of the equation by \frac{1}{3200}

(1000000 (x^{'})^{2} - 1000000 (y^{'})^{2}) \times \frac{1}{3200} = 3200 \times \frac{1}{3200}

Multiply the terms

More Steps

\frac{625}{2} (x^{'})^{2} - \frac{625}{2} (y^{'})^{2} = 3200 \times \frac{1}{3200}

Multiply the terms

More Steps

\frac{625}{2} (x^{'})^{2} - \frac{625}{2} (y^{'})^{2} = 1

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

\frac{( x ^{'} ) ^{2}}{\frac{2}{625}} - \frac{625}{2} (y^{'})^{2} = 1

Solution

\frac{( x ^{'} ) ^{2}}{\frac{2}{625}} - \frac{( y ^{'} ) ^{2}}{\frac{2}{625}} = 1

Show Solution