Solve x^2 + y^2 - 2 x - 4 y + 1 = 0

Question

Identify the conic

Find the standard equation of the circle

Find the radius of the circle

Find the center of the circle

(x - 1)^{2} + (y - 2)^{2} = 4

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Solve the equation

Solve for x

Solve for y

x = 1 + - y^{2} + 4 y x = 1 - - y^{2} + 4 y

Evaluate

x^{2} + y^{2} - 2 x - 4 y + 1 = 0

Rewrite the expression

x^{2} + y^{2} - 4 y + 1 - 2 x = 0

Rewrite in standard form

x^{2} - 2 x + y^{2} - 4 y + 1 = 0

Substitute a= 1,b= - 2 and c= y^{2} - 4 y + 1 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{2 \pm ( - 2 ) ^{2} - 4 ( y ^{2} - 4 y + 1 )}{2}

Simplify the expression

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x = \frac{2 \pm - 4 y ^{2} + 16 y}{2}

Simplify the radical expression

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x = \frac{2 \pm 2 - y ^{2} + 4 y}{2}

Separate the equation into 2 possible cases

x = \frac{2 + 2 - y ^{2} + 4 y}{2} x = \frac{2 - 2 - y ^{2} + 4 y}{2}

Simplify the expression

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x = 1 + - y^{2} + 4 y x = \frac{2 - 2 - y ^{2} + 4 y}{2}

Solution

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x = 1 + - y^{2} + 4 y x = 1 - - y^{2} + 4 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 + 1}{y - 2}

Calculate

x^{2} + y^{2} - 2 x - 4 y + 1 = 0

Take the derivative of both sides

\frac{d}{d x} (x^{2} + y^{2} - 2 x - 4 y + 1) = \frac{d}{d x} (0)

Calculate the derivative

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2 x + 2 y \frac{d y}{d x} - 2 - 4 \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

2 x + 2 y \frac{d y}{d x} - 2 - 4 \frac{d y}{d x} = 0

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

2 x - 2 + (2 y - 4) \frac{d y}{d x} = 0

Move the constant to the right side

(2 y - 4) \frac{d y}{d x} = 0 - (2 x - 2)

Subtract the terms

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(2 y - 4) \frac{d y}{d x} = - 2 x + 2

Divide both sides

\frac{( 2 y - 4 ) \frac{d y}{d x}}{2 y - 4} = \frac{- 2 x + 2}{2 y - 4}

Divide the numbers

\frac{d y}{d x} = \frac{- 2 x + 2}{2 y - 4}

Solution

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\frac{d y}{d x} = \frac{- x + 1}{y - 2}

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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} + 4 y - 5 - x ^{2} + 2 x}{y ^{3} - 6 y ^{2} + 12 y - 8}

Calculate

x^{2} + y^{2} - 2 x - 4 y + 1 = 0

Take the derivative of both sides

\frac{d}{d x} (x^{2} + y^{2} - 2 x - 4 y + 1) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

2 x + 2 y \frac{d y}{d x} - 2 - 4 \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

2 x + 2 y \frac{d y}{d x} - 2 - 4 \frac{d y}{d x} = 0

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

2 x - 2 + (2 y - 4) \frac{d y}{d x} = 0

Move the constant to the right side

(2 y - 4) \frac{d y}{d x} = 0 - (2 x - 2)

Subtract the terms

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(2 y - 4) \frac{d y}{d x} = - 2 x + 2

Divide both sides

\frac{( 2 y - 4 ) \frac{d y}{d x}}{2 y - 4} = \frac{- 2 x + 2}{2 y - 4}

Divide the numbers

\frac{d y}{d x} = \frac{- 2 x + 2}{2 y - 4}

Divide the numbers

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\frac{d y}{d x} = \frac{- x + 1}{y - 2}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{- x + 1}{y - 2})

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

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

Calculate the derivative

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

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

Calculate

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

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

Use equation \frac{d y}{d x} = \frac{- x + 1}{y - 2} to substitute

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- y ^{2} + 4 y - 5 - x ^{2} + 2 x}{y ^{3} - 6 y ^{2} + 12 y - 8}

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Rewrite the equation

r = cos (θ) + 2 sin (θ) + 3 sin^{2} (θ) + 2 sin (2 θ) r = cos (θ) + 2 sin (θ) - 3 sin^{2} (θ) + 2 sin (2 θ)

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Identify the conic

Solve the equation

Testing for symmetry

Find the first derivative

Find the second derivative

Rewrite the equation

Graph