Solve 25y^2=1600-64x^2

Identify the conic

Find the standard equation of the ellipse

Find the center of the ellipse

Find the foci of the ellipse

\frac{x ^{2}}{25} + \frac{y ^{2}}{64} = 1

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Solve the equation

Solve for x

Solve for y

x = \frac{5 64 - y ^{2}}{8} x = - \frac{5 64 - y ^{2}}{8}

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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

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{64 x}{25 y}

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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{1600 y ^{2} + 4096 x ^{2}}{625 y ^{3}}

Calculate

25 y^{2} = 1600 - 64 x^{2}

Take the derivative of both sides

\frac{d}{d x} (25 y^{2}) = \frac{d}{d x} (1600 - 64 x^{2})

Calculate the derivative

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50 y \frac{d y}{d x} = \frac{d}{d x} (1600 - 64 x^{2})

Calculate the derivative

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50 y \frac{d y}{d x} = - 128 x

Divide both sides

\frac{50 y \frac{d y}{d x}}{50 y} = \frac{- 128 x}{50 y}

Divide the numbers

\frac{d y}{d x} = \frac{- 128 x}{50 y}

Divide the numbers

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\frac{d y}{d x} = - \frac{64 x}{25 y}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{64 x}{25 y})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{64 x}{25 y})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( 64 x ) \times 25 y - 64 x \times \frac{d}{d x} ( 25 y )}{( 25 y ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = - \frac{64 \times 25 y - 64 x \times \frac{d}{d x} ( 25 y )}{( 25 y ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = - \frac{64 \times 25 y - 64 x \times 25 \frac{d y}{d x}}{( 25 y ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{1600 y - 64 x \times 25 \frac{d y}{d x}}{( 25 y ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{1600 y - 1600 x \frac{d y}{d x}}{( 25 y ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{1600 y - 1600 x \frac{d y}{d x}}{625 y ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{64 y - 64 x \frac{d y}{d x}}{25 y ^{2}}

Use equation \frac{d y}{d x} = - \frac{64 x}{25 y} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{64 y - 64 x ( - \frac{64 x}{25 y} )}{25 y ^{2}}

Solution

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\frac{d ^{2} y}{d x ^{2}} = - \frac{1600 y ^{2} + 4096 x ^{2}}{625 y ^{3}}

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Rewrite the equation

r = \frac{40 25 + 39 cos ^{2} ( θ )}{25 + 39 cos ^{2} ( θ )} r = - \frac{40 25 + 39 cos ^{2} ( θ )}{25 + 39 cos ^{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