Solve y^2 - 4*2 - 2 y - 48 x + 59 = 0

Identify the conic

Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

(y - 1)^{2} = 48 (x - \frac{25}{24})

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Solve the equation

Solve for x

Solve for y

x = \frac{y ^{2} + 51 - 2 y}{48}

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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{24}{y - 1}

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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{576}{y ^{3} - 3 y ^{2} + 3 y - 1}

Calculate

y^{2} - 4 \cdot 2 - 2 y - 48 x + 59 = 0

Simplify the expression

y^{2} + 51 - 2 y - 48 x = 0

Take the derivative of both sides

\frac{d}{d x} (y^{2} + 51 - 2 y - 48 x) = \frac{d}{d x} (0)

Calculate the derivative

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2 y \frac{d y}{d x} - 2 \frac{d y}{d x} - 48 = \frac{d}{d x} (0)

Calculate the derivative

2 y \frac{d y}{d x} - 2 \frac{d y}{d x} - 48 = 0

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

(2 y - 2) \frac{d y}{d x} - 48 = 0

Move the constant to the right side

(2 y - 2) \frac{d y}{d x} = 0 + 48

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

(2 y - 2) \frac{d y}{d x} = 48

Divide both sides

\frac{( 2 y - 2 ) \frac{d y}{d x}}{2 y - 2} = \frac{48}{2 y - 2}

Divide the numbers

\frac{d y}{d x} = \frac{48}{2 y - 2}

Divide the numbers

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\frac{d y}{d x} = \frac{24}{y - 1}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{24}{y - 1})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{24}{y - 1})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = 24 \times \frac{d}{d x} (\frac{1}{y - 1})

Rewrite the expression in exponential form

\frac{d ^{2} y}{d x ^{2}} = 24 \times \frac{d}{d x} ((y - 1)^{- 1})

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = 24 (- \frac{d y}{d x} \times (y - 1)^{- 2})

Rewrite the expression

\frac{d ^{2} y}{d x ^{2}} = 24 (- \frac{\frac{d y}{d x}}{( y - 1 ) ^{2}})

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{24 \frac{d y}{d x}}{( y - 1 ) ^{2}}

Use equation \frac{d y}{d x} = \frac{24}{y - 1} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{24 \times \frac{24}{y - 1}}{( y - 1 ) ^{2}}

Solution

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\frac{d ^{2} y}{d x ^{2}} = - \frac{576}{y ^{3} - 3 y ^{2} + 3 y - 1}

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Rewrite the equation

r = \frac{sin ( θ ) + 24 cos ( θ ) + - 50 + 626 cos ^{2} ( θ ) + 24 sin ( 2 θ )}{sin ^{2} ( θ )} r = \frac{sin ( θ ) + 24 cos ( θ ) - - 50 + 626 cos ^{2} ( θ ) + 24 sin ( 2 θ )}{sin ^{2} ( θ )}

Evaluate

y^{2} - 4 \times 2 - 2 y - 48 x + 59 = 0

Evaluate

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y^{2} + 51 - 2 y - 48 x = 0

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

(sin (θ) \times r)^{2} + 51 - 2 sin (θ) \times r - 48 cos (θ) \times r = 0

Factor the expression

sin^{2} (θ) \times r^{2} + (- 2 sin (θ) - 48 cos (θ)) r + 51 = 0

Solve using the quadratic formula

r = \frac{2 sin ( θ ) + 48 cos ( θ ) \pm ( - 2 sin ( θ ) - 48 cos ( θ ) ) ^{2} - 4 sin ^{2} ( θ ) \times 51}{2 sin ^{2} ( θ )}

Simplify

r = \frac{2 sin ( θ ) + 48 cos ( θ ) \pm - 200 + 2504 cos ^{2} ( θ ) + 96 sin ( 2 θ )}{2 sin ^{2} ( θ )}

Separate the equation into 2 possible cases

r = \frac{2 sin ( θ ) + 48 cos ( θ ) + - 200 + 2504 cos ^{2} ( θ ) + 96 sin ( 2 θ )}{2 sin ^{2} ( θ )} r = \frac{2 sin ( θ ) + 48 cos ( θ ) - - 200 + 2504 cos ^{2} ( θ ) + 96 sin ( 2 θ )}{2 sin ^{2} ( θ )}

Evaluate

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r = \frac{sin ( θ ) + 24 cos ( θ ) + - 50 + 626 cos ^{2} ( θ ) + 24 sin ( 2 θ )}{sin ^{2} ( θ )} r = \frac{2 sin ( θ ) + 48 cos ( θ ) - - 200 + 2504 cos ^{2} ( θ ) + 96 sin ( 2 θ )}{2 sin ^{2} ( θ )}

Solution

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r = \frac{sin ( θ ) + 24 cos ( θ ) + - 50 + 626 cos ^{2} ( θ ) + 24 sin ( 2 θ )}{sin ^{2} ( θ )} r = \frac{sin ( θ ) + 24 cos ( θ ) - - 50 + 626 cos ^{2} ( θ ) + 24 sin ( 2 θ )}{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

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