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Question
8x\times 2y=-10
Solve the equation
-
\text{Solve for }x
-
\text{Solve for }y
x=-\frac{5}{8y}
Evaluate
8x\times 2y=-10
Multiply the terms
16xy=-10
Rewrite the expression
16yx=-10
Divide both sides
\frac{16yx}{16y}=\frac{-10}{16y}
Divide the numbers
x=\frac{-10}{16y}
Solution
More Steps
Evaluate
\frac{-10}{16y}
\text{Cancel out the common factor }2
\frac{-5}{8y}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
-\frac{5}{8y}
x=-\frac{5}{8y}
Show Solutions
Testing for symmetry
-
Testing for symmetry about the origin
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Testing for symmetry about the x-axis
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Testing for symmetry about the y-axis
\textrm{Symmetry with respect to the origin}
Evaluate
8x\times 2y=-10
Multiply the terms
16xy=-10
\text{To test if the graph of }16xy=-10\text{ is symmetry with respect to the origin,substitute -x for x and -y for y}
16\left(-x\right)\left(-y\right)=-10
Evaluate
16xy=-10
Solution
\textrm{Symmetry with respect to the origin}
Show Solutions
Rewrite the equation
\begin{align}&r=\frac{\sqrt{5}}{2\sqrt{-\sin\left(2\theta \right)}}\\&r=-\frac{\sqrt{5}}{2\sqrt{-\sin\left(2\theta \right)}}\end{align}
Evaluate
8x\times 2y=-10
Evaluate
16xy=-10
\text{To convert the equation to polar coordinates,substitute }r\cos\left(\theta \right)\text{ for }x\text{ and }r\sin\left(\theta \right)\text{ for }y
16\cos\left(\theta \right)\times r\sin\left(\theta \right)\times r=-10
Factor the expression
16\cos\left(\theta \right)\sin\left(\theta \right)\times r^{2}=-10
Simplify the expression
8\sin\left(2\theta \right)\times r^{2}=-10
Divide the terms
r^{2}=-\frac{5}{4\sin\left(2\theta \right)}
Evaluate the power
r=\pm \sqrt{-\frac{5}{4\sin\left(2\theta \right)}}
Simplify the expression
More Steps
Evaluate
\sqrt{-\frac{5}{4\sin\left(2\theta \right)}}
To take a root of a fraction,take the root of the numerator and denominator separately
\frac{\sqrt{5}}{\sqrt{-4\sin\left(2\theta \right)}}
Simplify the radical expression
More Steps
Evaluate
\sqrt{-4\sin\left(2\theta \right)}
Write the expression as a product where the root of one of the factors can be evaluated
\sqrt{4\left(-\sin\left(2\theta \right)\right)}
\text{Write the number in exponential form with the base of }2
\sqrt{2^{2}\left(-\sin\left(2\theta \right)\right)}
Calculate
2\sqrt{-\sin\left(2\theta \right)}
\frac{\sqrt{5}}{2\sqrt{-\sin\left(2\theta \right)}}
r=\pm \frac{\sqrt{5}}{2\sqrt{-\sin\left(2\theta \right)}}
Solution
\begin{align}&r=\frac{\sqrt{5}}{2\sqrt{-\sin\left(2\theta \right)}}\\&r=-\frac{\sqrt{5}}{2\sqrt{-\sin\left(2\theta \right)}}\end{align}
Show Solutions
Find the first derivative
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\text{Find the derivative with respect to }x
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\text{Find the derivative with respect to }y
\frac{dy}{dx}=-\frac{y}{x}
Calculate
8x2y=-10
Simplify the expression
16xy=-10
Take the derivative of both sides
\frac{d}{dx}\left(16xy\right)=\frac{d}{dx}\left(-10\right)
Calculate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(16xy\right)
Use differentiation rules
\frac{d}{dx}\left(16x\right)\times y+16x\times \frac{d}{dx}\left(y\right)
Evaluate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(16x\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
16\times \frac{d}{dx}\left(x\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
16\times 1
Any expression multiplied by 1 remains the same
16
16y+16x\times \frac{d}{dx}\left(y\right)
Evaluate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(y\right)
Use differentiation rules
\frac{d}{dy}\left(y\right)\times \frac{dy}{dx}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
\frac{dy}{dx}
16y+16x\frac{dy}{dx}
16y+16x\frac{dy}{dx}=\frac{d}{dx}\left(-10\right)
Calculate the derivative
16y+16x\frac{dy}{dx}=0
Move the expression to the right-hand side and change its sign
16x\frac{dy}{dx}=0-16y
Removing 0 doesn't change the value,so remove it from the expression
16x\frac{dy}{dx}=-16y
Divide both sides
\frac{16x\frac{dy}{dx}}{16x}=\frac{-16y}{16x}
Divide the numbers
\frac{dy}{dx}=\frac{-16y}{16x}
Solution
More Steps
Evaluate
\frac{-16y}{16x}
\text{Cancel out the common factor }16
\frac{-y}{x}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
-\frac{y}{x}
\frac{dy}{dx}=-\frac{y}{x}
Show Solutions
Find the second derivative
-
\text{Find the second derivative with respect to }x
-
\text{Find the second derivative with respect to }y
\frac{d^2y}{dx^2}=\frac{2y}{x^{2}}
Calculate
8x2y=-10
Simplify the expression
16xy=-10
Take the derivative of both sides
\frac{d}{dx}\left(16xy\right)=\frac{d}{dx}\left(-10\right)
Calculate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(16xy\right)
Use differentiation rules
\frac{d}{dx}\left(16x\right)\times y+16x\times \frac{d}{dx}\left(y\right)
Evaluate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(16x\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
16\times \frac{d}{dx}\left(x\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
16\times 1
Any expression multiplied by 1 remains the same
16
16y+16x\times \frac{d}{dx}\left(y\right)
Evaluate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(y\right)
Use differentiation rules
\frac{d}{dy}\left(y\right)\times \frac{dy}{dx}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
\frac{dy}{dx}
16y+16x\frac{dy}{dx}
16y+16x\frac{dy}{dx}=\frac{d}{dx}\left(-10\right)
Calculate the derivative
16y+16x\frac{dy}{dx}=0
Move the expression to the right-hand side and change its sign
16x\frac{dy}{dx}=0-16y
Removing 0 doesn't change the value,so remove it from the expression
16x\frac{dy}{dx}=-16y
Divide both sides
\frac{16x\frac{dy}{dx}}{16x}=\frac{-16y}{16x}
Divide the numbers
\frac{dy}{dx}=\frac{-16y}{16x}
Divide the numbers
More Steps
Evaluate
\frac{-16y}{16x}
\text{Cancel out the common factor }16
\frac{-y}{x}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
-\frac{y}{x}
\frac{dy}{dx}=-\frac{y}{x}
Take the derivative of both sides
\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(-\frac{y}{x}\right)
Calculate the derivative
\frac{d^2y}{dx^2}=\frac{d}{dx}\left(-\frac{y}{x}\right)
Use differentiation rules
\frac{d^2y}{dx^2}=-\frac{\frac{d}{dx}\left(y\right)\times x-y\times \frac{d}{dx}\left(x\right)}{x^{2}}
Calculate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(y\right)
Use differentiation rules
\frac{d}{dy}\left(y\right)\times \frac{dy}{dx}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
\frac{dy}{dx}
\frac{d^2y}{dx^2}=-\frac{\frac{dy}{dx}\times x-y\times \frac{d}{dx}\left(x\right)}{x^{2}}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
\frac{d^2y}{dx^2}=-\frac{\frac{dy}{dx}\times x-y\times 1}{x^{2}}
Use the commutative property to reorder the terms
\frac{d^2y}{dx^2}=-\frac{x\frac{dy}{dx}-y\times 1}{x^{2}}
Any expression multiplied by 1 remains the same
\frac{d^2y}{dx^2}=-\frac{x\frac{dy}{dx}-y}{x^{2}}
\text{Use equation }\frac{dy}{dx}=-\frac{y}{x}\text{ to substitute}
\frac{d^2y}{dx^2}=-\frac{x\left(-\frac{y}{x}\right)-y}{x^{2}}
Solution
More Steps
Calculate
-\frac{x\left(-\frac{y}{x}\right)-y}{x^{2}}
Multiply the terms
More Steps
Evaluate
x\left(-\frac{y}{x}\right)
Multiplying or dividing an odd number of negative terms equals a negative
-x\times \frac{y}{x}
\text{Cancel out the common factor }x
-1\times y
Multiply the terms
-y
-\frac{-y-y}{x^{2}}
Subtract the terms
More Steps
Simplify
-y-y
Collect like terms by calculating the sum or difference of their coefficients
\left(-1-1\right)y
Subtract the numbers
-2y
-\frac{-2y}{x^{2}}
Divide the terms
-\left(-\frac{2y}{x^{2}}\right)
Calculate
\frac{2y}{x^{2}}
\frac{d^2y}{dx^2}=\frac{2y}{x^{2}}
Show Solutions
Conic
\frac{\left(y^{\prime}\right)^{2}}{\frac{5}{4}}-\frac{\left(x^{\prime}\right)^{2}}{\frac{5}{4}}=1
Evaluate
8x\times 2y=-10
Move the expression to the left side
8x\times 2y-\left(-10\right)=0
Calculate
More Steps
Calculate
8x\times 2y-\left(-10\right)
Multiply the terms
16xy-\left(-10\right)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
16xy+10
16xy+10=0
\text{The coefficients A,B and C of the general equation are A=}0\text{,B=}16\text{ and C=}0
\begin{align}&A=0\\&B=16\\&C=0\end{align}
\text{To find the angle of rotation }\theta\text{,substitute the values of A,B and C into the formula }\cot(2\theta)=\frac{A-C}{B}
\cot\left(2\theta \right)=\frac{0-0}{16}
Calculate
\cot\left(2\theta \right)=0
\text{Using the unit circle,find the smallest positive angle for which the cotangent is }0
2\theta =\frac{\pi }{2}
Calculate
\theta =\frac{\pi }{4}
\text{To rotate the axes,use the equation of rotation and substitute }\frac{\pi }{4}\text{ for }\theta
\begin{align}&x=x^{\prime}\cos\left(\frac{\pi }{4}\right)-y^{\prime}\sin\left(\frac{\pi }{4}\right)\\&y=x^{\prime}\sin\left(\frac{\pi }{4}\right)+y^{\prime}\cos\left(\frac{\pi }{4}\right)\end{align}
Calculate
\begin{align}&x=x^{\prime}\times \frac{\sqrt{2}}{2}-y^{\prime}\sin\left(\frac{\pi }{4}\right)\\&y=x^{\prime}\sin\left(\frac{\pi }{4}\right)+y^{\prime}\cos\left(\frac{\pi }{4}\right)\end{align}
Calculate
\begin{align}&x=x^{\prime}\times \frac{\sqrt{2}}{2}-y^{\prime}\times \frac{\sqrt{2}}{2}\\&y=x^{\prime}\sin\left(\frac{\pi }{4}\right)+y^{\prime}\cos\left(\frac{\pi }{4}\right)\end{align}
Calculate
\begin{align}&x=x^{\prime}\times \frac{\sqrt{2}}{2}-y^{\prime}\times \frac{\sqrt{2}}{2}\\&y=x^{\prime}\times \frac{\sqrt{2}}{2}+y^{\prime}\cos\left(\frac{\pi }{4}\right)\end{align}
Calculate
\begin{align}&x=x^{\prime}\times \frac{\sqrt{2}}{2}-y^{\prime}\times \frac{\sqrt{2}}{2}\\&y=x^{\prime}\times \frac{\sqrt{2}}{2}+y^{\prime}\times \frac{\sqrt{2}}{2}\end{align}
\text{Substitute x and y into the original equation }16xy+10=0
16\left(x^{\prime}\times \frac{\sqrt{2}}{2}-y^{\prime}\times \frac{\sqrt{2}}{2}\right)\left(x^{\prime}\times \frac{\sqrt{2}}{2}+y^{\prime}\times \frac{\sqrt{2}}{2}\right)+10=0
Calculate
More Steps
Calculate
16\left(x^{\prime}\times \frac{\sqrt{2}}{2}-y^{\prime}\times \frac{\sqrt{2}}{2}\right)\left(x^{\prime}\times \frac{\sqrt{2}}{2}+y^{\prime}\times \frac{\sqrt{2}}{2}\right)+10
Use the commutative property to reorder the terms
16\left(\frac{\sqrt{2}}{2}x^{\prime}-y^{\prime}\times \frac{\sqrt{2}}{2}\right)\left(x^{\prime}\times \frac{\sqrt{2}}{2}+y^{\prime}\times \frac{\sqrt{2}}{2}\right)+10
Use the commutative property to reorder the terms
16\left(\frac{\sqrt{2}}{2}x^{\prime}-\frac{\sqrt{2}}{2}y^{\prime}\right)\left(x^{\prime}\times \frac{\sqrt{2}}{2}+y^{\prime}\times \frac{\sqrt{2}}{2}\right)+10
Use the commutative property to reorder the terms
16\left(\frac{\sqrt{2}}{2}x^{\prime}-\frac{\sqrt{2}}{2}y^{\prime}\right)\left(\frac{\sqrt{2}}{2}x^{\prime}+y^{\prime}\times \frac{\sqrt{2}}{2}\right)+10
Use the commutative property to reorder the terms
16\left(\frac{\sqrt{2}}{2}x^{\prime}-\frac{\sqrt{2}}{2}y^{\prime}\right)\left(\frac{\sqrt{2}}{2}x^{\prime}+\frac{\sqrt{2}}{2}y^{\prime}\right)+10
Expand the expression
More Steps
Calculate
16\left(\frac{\sqrt{2}}{2}x^{\prime}-\frac{\sqrt{2}}{2}y^{\prime}\right)\left(\frac{\sqrt{2}}{2}x^{\prime}+\frac{\sqrt{2}}{2}y^{\prime}\right)
Simplify
\left(8\sqrt{2}\times x^{\prime}-8\sqrt{2}\times y^{\prime}\right)\left(\frac{\sqrt{2}}{2}x^{\prime}+\frac{\sqrt{2}}{2}y^{\prime}\right)
Apply the distributive property
8\sqrt{2}\times x^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}+8\sqrt{2}\times x^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}-8\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}-8\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the terms
8\left(x^{\prime}\right)^{2}+8\sqrt{2}\times x^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}-8\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}-8\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the numbers
8\left(x^{\prime}\right)^{2}+8x^{\prime}y^{\prime}-8\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}-8\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the numbers
8\left(x^{\prime}\right)^{2}+8x^{\prime}y^{\prime}-8y^{\prime}x^{\prime}-8\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the terms
8\left(x^{\prime}\right)^{2}+8x^{\prime}y^{\prime}-8y^{\prime}x^{\prime}-8\left(y^{\prime}\right)^{2}
Subtract the terms
8\left(x^{\prime}\right)^{2}+0-8\left(y^{\prime}\right)^{2}
Removing 0 doesn't change the value,so remove it from the expression
8\left(x^{\prime}\right)^{2}-8\left(y^{\prime}\right)^{2}
8\left(x^{\prime}\right)^{2}-8\left(y^{\prime}\right)^{2}+10
8\left(x^{\prime}\right)^{2}-8\left(y^{\prime}\right)^{2}+10=0
Move the constant to the right-hand side and change its sign
8\left(x^{\prime}\right)^{2}-8\left(y^{\prime}\right)^{2}=0-10
Removing 0 doesn't change the value,so remove it from the expression
8\left(x^{\prime}\right)^{2}-8\left(y^{\prime}\right)^{2}=-10
\text{Multiply both sides of the equation by }-\frac{1}{10}
\left(8\left(x^{\prime}\right)^{2}-8\left(y^{\prime}\right)^{2}\right)\left(-\frac{1}{10}\right)=-10\left(-\frac{1}{10}\right)
Multiply the terms
More Steps
Evaluate
\left(8\left(x^{\prime}\right)^{2}-8\left(y^{\prime}\right)^{2}\right)\left(-\frac{1}{10}\right)
Use the the distributive property to expand the expression
8\left(x^{\prime}\right)^{2}\left(-\frac{1}{10}\right)-8\left(y^{\prime}\right)^{2}\left(-\frac{1}{10}\right)
Multiply the numbers
More Steps
Evaluate
8\left(-\frac{1}{10}\right)
Multiplying or dividing an odd number of negative terms equals a negative
-8\times \frac{1}{10}
Reduce the numbers
-4\times \frac{1}{5}
Multiply the numbers
-\frac{4}{5}
-\frac{4}{5}\left(x^{\prime}\right)^{2}-8\left(y^{\prime}\right)^{2}\left(-\frac{1}{10}\right)
Multiply the numbers
More Steps
Evaluate
-8\left(-\frac{1}{10}\right)
Multiplying or dividing an even number of negative terms equals a positive
8\times \frac{1}{10}
Reduce the numbers
4\times \frac{1}{5}
Multiply the numbers
\frac{4}{5}
-\frac{4}{5}\left(x^{\prime}\right)^{2}+\frac{4}{5}\left(y^{\prime}\right)^{2}
-\frac{4}{5}\left(x^{\prime}\right)^{2}+\frac{4}{5}\left(y^{\prime}\right)^{2}=-10\left(-\frac{1}{10}\right)
Multiply the terms
More Steps
Evaluate
-10\left(-\frac{1}{10}\right)
Multiplying or dividing an even number of negative terms equals a positive
10\times \frac{1}{10}
Reduce the numbers
1\times 1
Simplify
1
-\frac{4}{5}\left(x^{\prime}\right)^{2}+\frac{4}{5}\left(y^{\prime}\right)^{2}=1
\text{Use }a = \frac{1}{ \frac{1}{a} }\text{ to transform the expression}
-\frac{\left(x^{\prime}\right)^{2}}{\frac{5}{4}}+\frac{4}{5}\left(y^{\prime}\right)^{2}=1
\text{Use }a = \frac{1}{ \frac{1}{a} }\text{ to transform the expression}
-\frac{\left(x^{\prime}\right)^{2}}{\frac{5}{4}}+\frac{\left(y^{\prime}\right)^{2}}{\frac{5}{4}}=1
Solution
\frac{\left(y^{\prime}\right)^{2}}{\frac{5}{4}}-\frac{\left(x^{\prime}\right)^{2}}{\frac{5}{4}}=1
Show Solutions
