Solve \frac{(x 1)^2}{16} \frac{(y-2)^2}{4} = 1

Question

Solve the equation

Solve for x

Solve for y

x = \frac{8}{∣ y - 2 ∣} x = - \frac{8}{∣ y - 2 ∣}

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

Calculate

\frac{( x 1 ) ^{2}}{16} \frac{( y - 2 ) ^{2}}{4} = 1

Simplify the expression

\frac{x ^{2} ( y - 2 ) ^{2}}{64} = 1

Take the derivative of both sides

\frac{d}{d x} (\frac{x ^{2} ( y - 2 ) ^{2}}{64}) = \frac{d}{d x} (1)

Calculate the derivative

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\frac{x ( y - 2 ) ^{2} + x ^{2} \frac{d y}{d x} \times ( y - 2 )}{32} = \frac{d}{d x} (1)

Calculate the derivative

\frac{x ( y - 2 ) ^{2} + x ^{2} \frac{d y}{d x} \times ( y - 2 )}{32} = 0

Rewrite the expression

\frac{( y - 2 ) ^{2} x + ( x ^{2} y - 2 x ^{2} ) \frac{d y}{d x}}{32} = 0

Simplify

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

Move the constant to the right side

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

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

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

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

Expand the expression

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

Simplify

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

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

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

Solution

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

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