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Question
3x\times 2y=8
Solve the equation
-
\text{Solve for }x
-
\text{Solve for }y
x=\frac{4}{3y}
Evaluate
3x\times 2y=8
Multiply the terms
6xy=8
Rewrite the expression
6yx=8
Divide both sides
\frac{6yx}{6y}=\frac{8}{6y}
Divide the numbers
x=\frac{8}{6y}
Solution
x=\frac{4}{3y}
Show Solutions
Testing for symmetry
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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
3x\times 2y=8
Multiply the terms
6xy=8
\text{To test if the graph of }6xy=8\text{ is symmetry with respect to the origin,substitute -x for x and -y for y}
6\left(-x\right)\left(-y\right)=8
Evaluate
6xy=8
Solution
\textrm{Symmetry with respect to the origin}
Show Solutions
Rewrite the equation
\begin{align}&r=\frac{2\sqrt{6\sin\left(2\theta \right)}}{3\left|\sin\left(2\theta \right)\right|}\\&r=-\frac{2\sqrt{6\sin\left(2\theta \right)}}{3\left|\sin\left(2\theta \right)\right|}\end{align}
Evaluate
3x\times 2y=8
Evaluate
6xy=8
\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
6\cos\left(\theta \right)\times r\sin\left(\theta \right)\times r=8
Factor the expression
6\cos\left(\theta \right)\sin\left(\theta \right)\times r^{2}=8
Simplify the expression
3\sin\left(2\theta \right)\times r^{2}=8
Divide the terms
r^{2}=\frac{8}{3\sin\left(2\theta \right)}
Evaluate the power
r=\pm \sqrt{\frac{8}{3\sin\left(2\theta \right)}}
Simplify the expression
More Steps
Evaluate
\sqrt{\frac{8}{3\sin\left(2\theta \right)}}
To take a root of a fraction,take the root of the numerator and denominator separately
\frac{\sqrt{8}}{\sqrt{3\sin\left(2\theta \right)}}
Simplify the radical expression
More Steps
Evaluate
\sqrt{8}
Write the expression as a product where the root of one of the factors can be evaluated
\sqrt{4\times 2}
\text{Write the number in exponential form with the base of }2
\sqrt{2^{2}\times 2}
The root of a product is equal to the product of the roots of each factor
\sqrt{2^{2}}\times \sqrt{2}
\text{Reduce the index of the radical and exponent with }2
2\sqrt{2}
\frac{2\sqrt{2}}{\sqrt{3\sin\left(2\theta \right)}}
Multiply by the Conjugate
\frac{2\sqrt{2}\times \sqrt{3\sin\left(2\theta \right)}}{\sqrt{3\sin\left(2\theta \right)}\times \sqrt{3\sin\left(2\theta \right)}}
Calculate
\frac{2\sqrt{2}\times \sqrt{3\sin\left(2\theta \right)}}{3\left|\sin\left(2\theta \right)\right|}
Calculate the product
More Steps
Evaluate
\sqrt{2}\times \sqrt{3\sin\left(2\theta \right)}
The product of roots with the same index is equal to the root of the product
\sqrt{2\times 3\sin\left(2\theta \right)}
Calculate the product
\sqrt{6\sin\left(2\theta \right)}
\frac{2\sqrt{6\sin\left(2\theta \right)}}{3\left|\sin\left(2\theta \right)\right|}
r=\pm \frac{2\sqrt{6\sin\left(2\theta \right)}}{3\left|\sin\left(2\theta \right)\right|}
Solution
\begin{align}&r=\frac{2\sqrt{6\sin\left(2\theta \right)}}{3\left|\sin\left(2\theta \right)\right|}\\&r=-\frac{2\sqrt{6\sin\left(2\theta \right)}}{3\left|\sin\left(2\theta \right)\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
3x2y=8
Simplify the expression
6xy=8
Take the derivative of both sides
\frac{d}{dx}\left(6xy\right)=\frac{d}{dx}\left(8\right)
Calculate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(6xy\right)
Use differentiation rules
\frac{d}{dx}\left(6x\right)\times y+6x\times \frac{d}{dx}\left(y\right)
Evaluate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(6x\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
6\times \frac{d}{dx}\left(x\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
6\times 1
Any expression multiplied by 1 remains the same
6
6y+6x\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}
6y+6x\frac{dy}{dx}
6y+6x\frac{dy}{dx}=\frac{d}{dx}\left(8\right)
Calculate the derivative
6y+6x\frac{dy}{dx}=0
Move the expression to the right-hand side and change its sign
6x\frac{dy}{dx}=0-6y
Removing 0 doesn't change the value,so remove it from the expression
6x\frac{dy}{dx}=-6y
Divide both sides
\frac{6x\frac{dy}{dx}}{6x}=\frac{-6y}{6x}
Divide the numbers
\frac{dy}{dx}=\frac{-6y}{6x}
Solution
More Steps
Evaluate
\frac{-6y}{6x}
\text{Cancel out the common factor }6
\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
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\text{Find the second derivative with respect to }x
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\text{Find the second derivative with respect to }y
\frac{d^2y}{dx^2}=\frac{2y}{x^{2}}
Calculate
3x2y=8
Simplify the expression
6xy=8
Take the derivative of both sides
\frac{d}{dx}\left(6xy\right)=\frac{d}{dx}\left(8\right)
Calculate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(6xy\right)
Use differentiation rules
\frac{d}{dx}\left(6x\right)\times y+6x\times \frac{d}{dx}\left(y\right)
Evaluate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(6x\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
6\times \frac{d}{dx}\left(x\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
6\times 1
Any expression multiplied by 1 remains the same
6
6y+6x\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}
6y+6x\frac{dy}{dx}
6y+6x\frac{dy}{dx}=\frac{d}{dx}\left(8\right)
Calculate the derivative
6y+6x\frac{dy}{dx}=0
Move the expression to the right-hand side and change its sign
6x\frac{dy}{dx}=0-6y
Removing 0 doesn't change the value,so remove it from the expression
6x\frac{dy}{dx}=-6y
Divide both sides
\frac{6x\frac{dy}{dx}}{6x}=\frac{-6y}{6x}
Divide the numbers
\frac{dy}{dx}=\frac{-6y}{6x}
Divide the numbers
More Steps
Evaluate
\frac{-6y}{6x}
\text{Cancel out the common factor }6
\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(x^{\prime}\right)^{2}}{\frac{8}{3}}-\frac{\left(y^{\prime}\right)^{2}}{\frac{8}{3}}=1
Evaluate
3x\times 2y=8
Move the expression to the left side
3x\times 2y-8=0
Calculate
6xy-8=0
\text{The coefficients A,B and C of the general equation are A=}0\text{,B=}6\text{ and C=}0
\begin{align}&A=0\\&B=6\\&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}{6}
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 }6xy-8=0
6\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)-8=0
Calculate
More Steps
Calculate
6\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)-8
Use the commutative property to reorder the terms
6\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)-8
Use the commutative property to reorder the terms
6\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)-8
Use the commutative property to reorder the terms
6\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)-8
Use the commutative property to reorder the terms
6\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)-8
Expand the expression
More Steps
Calculate
6\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(3\sqrt{2}\times x^{\prime}-3\sqrt{2}\times y^{\prime}\right)\left(\frac{\sqrt{2}}{2}x^{\prime}+\frac{\sqrt{2}}{2}y^{\prime}\right)
Apply the distributive property
3\sqrt{2}\times x^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}+3\sqrt{2}\times x^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}-3\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}-3\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the terms
3\left(x^{\prime}\right)^{2}+3\sqrt{2}\times x^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}-3\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}-3\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the numbers
3\left(x^{\prime}\right)^{2}+3x^{\prime}y^{\prime}-3\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}-3\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the numbers
3\left(x^{\prime}\right)^{2}+3x^{\prime}y^{\prime}-3y^{\prime}x^{\prime}-3\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the terms
3\left(x^{\prime}\right)^{2}+3x^{\prime}y^{\prime}-3y^{\prime}x^{\prime}-3\left(y^{\prime}\right)^{2}
Subtract the terms
3\left(x^{\prime}\right)^{2}+0-3\left(y^{\prime}\right)^{2}
Removing 0 doesn't change the value,so remove it from the expression
3\left(x^{\prime}\right)^{2}-3\left(y^{\prime}\right)^{2}
3\left(x^{\prime}\right)^{2}-3\left(y^{\prime}\right)^{2}-8
3\left(x^{\prime}\right)^{2}-3\left(y^{\prime}\right)^{2}-8=0
Move the constant to the right-hand side and change its sign
3\left(x^{\prime}\right)^{2}-3\left(y^{\prime}\right)^{2}=0-\left(-8\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
3\left(x^{\prime}\right)^{2}-3\left(y^{\prime}\right)^{2}=0+8
Removing 0 doesn't change the value,so remove it from the expression
3\left(x^{\prime}\right)^{2}-3\left(y^{\prime}\right)^{2}=8
\text{Multiply both sides of the equation by }\frac{1}{8}
\left(3\left(x^{\prime}\right)^{2}-3\left(y^{\prime}\right)^{2}\right)\times \frac{1}{8}=8\times \frac{1}{8}
Multiply the terms
More Steps
Evaluate
\left(3\left(x^{\prime}\right)^{2}-3\left(y^{\prime}\right)^{2}\right)\times \frac{1}{8}
Use the the distributive property to expand the expression
3\left(x^{\prime}\right)^{2}\times \frac{1}{8}-3\left(y^{\prime}\right)^{2}\times \frac{1}{8}
Multiply the numbers
\frac{3}{8}\left(x^{\prime}\right)^{2}-3\left(y^{\prime}\right)^{2}\times \frac{1}{8}
Multiply the numbers
\frac{3}{8}\left(x^{\prime}\right)^{2}-\frac{3}{8}\left(y^{\prime}\right)^{2}
\frac{3}{8}\left(x^{\prime}\right)^{2}-\frac{3}{8}\left(y^{\prime}\right)^{2}=8\times \frac{1}{8}
Multiply the terms
More Steps
Evaluate
8\times \frac{1}{8}
Reduce the numbers
1\times 1
Simplify
1
\frac{3}{8}\left(x^{\prime}\right)^{2}-\frac{3}{8}\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{8}{3}}-\frac{3}{8}\left(y^{\prime}\right)^{2}=1
Solution
\frac{\left(x^{\prime}\right)^{2}}{\frac{8}{3}}-\frac{\left(y^{\prime}\right)^{2}}{\frac{8}{3}}=1
Show Solutions
