Solve x^2 + y^2 - 6 y = 0

Identify the conic

Find the standard equation of the circle

Find the radius of the circle

Find the center of the circle

x^{2} + (y - 3)^{2} = 9

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Solve the equation

$Solve for x$

$Solve for y$

x = - y^{2} + 6 y x = - - y^{2} + 6 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

Not symmetry with respect to the origin

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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{x}{y - 3}

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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{y ^{2} - 6 y + 9 + x ^{2}}{y ^{3} - 9 y ^{2} + 27 y - 27}

Calculate

x^{2} + y^{2} - 6 y = 0

Take the derivative of both sides

\frac{d}{d x} (x^{2} + y^{2} - 6 y) = \frac{d}{d x} (0)

Calculate the derivative

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2 x + 2 y \frac{d y}{d x} - 6 \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

2 x + 2 y \frac{d y}{d x} - 6 \frac{d y}{d x} = 0

Collect like terms by calculating the sum or difference of their coefficients

2 x + (2 y - 6) \frac{d y}{d x} = 0

Move the constant to the right side

(2 y - 6) \frac{d y}{d x} = 0 - 2 x

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

(2 y - 6) \frac{d y}{d x} = - 2 x

Divide both sides

\frac{( 2 y - 6 ) \frac{d y}{d x}}{2 y - 6} = \frac{- 2 x}{2 y - 6}

Divide the numbers

\frac{d y}{d x} = \frac{- 2 x}{2 y - 6}

Divide the numbers

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

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{x}{y - 3})

Calculate the derivative

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

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( x ) \times ( y - 3 ) - x \times \frac{d}{d x} ( y - 3 )}{( y - 3 ) ^{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 ( y - 3 ) - x \times \frac{d}{d x} ( y - 3 )}{( y - 3 ) ^{2}}

Calculate the derivative

More Steps

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

Any expression multiplied by 1 remains the same

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

Use equation \frac{d y}{d x} = - \frac{x}{y - 3} to substitute

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{y ^{2} - 6 y + 9 + x ^{2}}{y ^{3} - 9 y ^{2} + 27 y - 27}

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Rewrite the equation

Rewrite in polar form

r = 0 r = 6 sin (θ)

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X^2 + y^2 6 y = 0

Identify the conic

Find the standard equation of the circle

Find the radius of the circle

Find the center of the circle

Solve the equation

$Solve for x$

$Solve for y$

Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Find the first derivative

$Find the derivative with respect to x$

$Find the derivative with respect to y$

Find the second derivative

$Find the second derivative with respect to x$

$Find the second derivative with respect to y$

Rewrite the equation

Rewrite in polar form

Graph