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Question
4x\times 8y=-48
Solve the equation
-
\text{Solve for }x
-
\text{Solve for }y
x=-\frac{3}{2y}
Evaluate
4x\times 8y=-48
Multiply the terms
32xy=-48
Rewrite the expression
32yx=-48
Divide both sides
\frac{32yx}{32y}=\frac{-48}{32y}
Divide the numbers
x=\frac{-48}{32y}
Solution
More Steps
Evaluate
\frac{-48}{32y}
\text{Cancel out the common factor }16
\frac{-3}{2y}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
-\frac{3}{2y}
x=-\frac{3}{2y}
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
4x\times 8y=-48
Multiply the terms
32xy=-48
\text{To test if the graph of }32xy=-48\text{ is symmetry with respect to the origin,substitute -x for x and -y for y}
32\left(-x\right)\left(-y\right)=-48
Evaluate
32xy=-48
Solution
\textrm{Symmetry with respect to the origin}
Show Solutions
Rewrite the equation
\begin{align}&r=\sqrt{-3\csc\left(2\theta \right)}\\&r=-\sqrt{-3\csc\left(2\theta \right)}\end{align}
Evaluate
4x\times 8y=-48
Evaluate
32xy=-48
\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
32\cos\left(\theta \right)\times r\sin\left(\theta \right)\times r=-48
Factor the expression
32\cos\left(\theta \right)\sin\left(\theta \right)\times r^{2}=-48
Simplify the expression
16\sin\left(2\theta \right)\times r^{2}=-48
Divide the terms
r^{2}=-\frac{3}{\sin\left(2\theta \right)}
Simplify the expression
r^{2}=-3\csc\left(2\theta \right)
Evaluate the power
r=\pm \sqrt{-3\csc\left(2\theta \right)}
Solution
\begin{align}&r=\sqrt{-3\csc\left(2\theta \right)}\\&r=-\sqrt{-3\csc\left(2\theta \right)}\end{align}
Show Solutions
Find the first derivative
-
\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
4x8y=-48
Simplify the expression
32xy=-48
Take the derivative of both sides
\frac{d}{dx}\left(32xy\right)=\frac{d}{dx}\left(-48\right)
Calculate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(32xy\right)
Use differentiation rules
\frac{d}{dx}\left(32x\right)\times y+32x\times \frac{d}{dx}\left(y\right)
Evaluate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(32x\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
32\times \frac{d}{dx}\left(x\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
32\times 1
Any expression multiplied by 1 remains the same
32
32y+32x\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}
32y+32x\frac{dy}{dx}
32y+32x\frac{dy}{dx}=\frac{d}{dx}\left(-48\right)
Calculate the derivative
32y+32x\frac{dy}{dx}=0
Move the expression to the right-hand side and change its sign
32x\frac{dy}{dx}=0-32y
Removing 0 doesn't change the value,so remove it from the expression
32x\frac{dy}{dx}=-32y
Divide both sides
\frac{32x\frac{dy}{dx}}{32x}=\frac{-32y}{32x}
Divide the numbers
\frac{dy}{dx}=\frac{-32y}{32x}
Solution
More Steps
Evaluate
\frac{-32y}{32x}
\text{Cancel out the common factor }32
\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
-
\text{Find the second derivative with respect to }y
\frac{d^2y}{dx^2}=\frac{2y}{x^{2}}
Calculate
4x8y=-48
Simplify the expression
32xy=-48
Take the derivative of both sides
\frac{d}{dx}\left(32xy\right)=\frac{d}{dx}\left(-48\right)
Calculate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(32xy\right)
Use differentiation rules
\frac{d}{dx}\left(32x\right)\times y+32x\times \frac{d}{dx}\left(y\right)
Evaluate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(32x\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
32\times \frac{d}{dx}\left(x\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
32\times 1
Any expression multiplied by 1 remains the same
32
32y+32x\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}
32y+32x\frac{dy}{dx}
32y+32x\frac{dy}{dx}=\frac{d}{dx}\left(-48\right)
Calculate the derivative
32y+32x\frac{dy}{dx}=0
Move the expression to the right-hand side and change its sign
32x\frac{dy}{dx}=0-32y
Removing 0 doesn't change the value,so remove it from the expression
32x\frac{dy}{dx}=-32y
Divide both sides
\frac{32x\frac{dy}{dx}}{32x}=\frac{-32y}{32x}
Divide the numbers
\frac{dy}{dx}=\frac{-32y}{32x}
Divide the numbers
More Steps
Evaluate
\frac{-32y}{32x}
\text{Cancel out the common factor }32
\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}}{3}-\frac{\left(x^{\prime}\right)^{2}}{3}=1
Evaluate
4x\times 8y=-48
Move the expression to the left side
4x\times 8y-\left(-48\right)=0
Calculate
More Steps
Calculate
4x\times 8y-\left(-48\right)
Multiply the terms
32xy-\left(-48\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
32xy+48
32xy+48=0
\text{The coefficients A,B and C of the general equation are A=}0\text{,B=}32\text{ and C=}0
\begin{align}&A=0\\&B=32\\&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}{32}
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 }32xy+48=0
32\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)+48=0
Calculate
More Steps
Calculate
32\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)+48
Use the commutative property to reorder the terms
32\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)+48
Use the commutative property to reorder the terms
32\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)+48
Use the commutative property to reorder the terms
32\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)+48
Use the commutative property to reorder the terms
32\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)+48
Expand the expression
More Steps
Calculate
32\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(16\sqrt{2}\times x^{\prime}-16\sqrt{2}\times y^{\prime}\right)\left(\frac{\sqrt{2}}{2}x^{\prime}+\frac{\sqrt{2}}{2}y^{\prime}\right)
Apply the distributive property
16\sqrt{2}\times x^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}+16\sqrt{2}\times x^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}-16\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}-16\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the terms
16\left(x^{\prime}\right)^{2}+16\sqrt{2}\times x^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}-16\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}-16\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the numbers
16\left(x^{\prime}\right)^{2}+16x^{\prime}y^{\prime}-16\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}-16\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the numbers
16\left(x^{\prime}\right)^{2}+16x^{\prime}y^{\prime}-16y^{\prime}x^{\prime}-16\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the terms
16\left(x^{\prime}\right)^{2}+16x^{\prime}y^{\prime}-16y^{\prime}x^{\prime}-16\left(y^{\prime}\right)^{2}
Subtract the terms
16\left(x^{\prime}\right)^{2}+0-16\left(y^{\prime}\right)^{2}
Removing 0 doesn't change the value,so remove it from the expression
16\left(x^{\prime}\right)^{2}-16\left(y^{\prime}\right)^{2}
16\left(x^{\prime}\right)^{2}-16\left(y^{\prime}\right)^{2}+48
16\left(x^{\prime}\right)^{2}-16\left(y^{\prime}\right)^{2}+48=0
Move the constant to the right-hand side and change its sign
16\left(x^{\prime}\right)^{2}-16\left(y^{\prime}\right)^{2}=0-48
Removing 0 doesn't change the value,so remove it from the expression
16\left(x^{\prime}\right)^{2}-16\left(y^{\prime}\right)^{2}=-48
\text{Multiply both sides of the equation by }-\frac{1}{48}
\left(16\left(x^{\prime}\right)^{2}-16\left(y^{\prime}\right)^{2}\right)\left(-\frac{1}{48}\right)=-48\left(-\frac{1}{48}\right)
Multiply the terms
More Steps
Evaluate
\left(16\left(x^{\prime}\right)^{2}-16\left(y^{\prime}\right)^{2}\right)\left(-\frac{1}{48}\right)
Use the the distributive property to expand the expression
16\left(x^{\prime}\right)^{2}\left(-\frac{1}{48}\right)-16\left(y^{\prime}\right)^{2}\left(-\frac{1}{48}\right)
Multiply the numbers
More Steps
Evaluate
16\left(-\frac{1}{48}\right)
Multiplying or dividing an odd number of negative terms equals a negative
-16\times \frac{1}{48}
Reduce the numbers
-1\times \frac{1}{3}
Multiply the numbers
-\frac{1}{3}
-\frac{1}{3}\left(x^{\prime}\right)^{2}-16\left(y^{\prime}\right)^{2}\left(-\frac{1}{48}\right)
Multiply the numbers
More Steps
Evaluate
-16\left(-\frac{1}{48}\right)
Multiplying or dividing an even number of negative terms equals a positive
16\times \frac{1}{48}
Reduce the numbers
1\times \frac{1}{3}
Multiply the numbers
\frac{1}{3}
-\frac{1}{3}\left(x^{\prime}\right)^{2}+\frac{1}{3}\left(y^{\prime}\right)^{2}
-\frac{1}{3}\left(x^{\prime}\right)^{2}+\frac{1}{3}\left(y^{\prime}\right)^{2}=-48\left(-\frac{1}{48}\right)
Multiply the terms
More Steps
Evaluate
-48\left(-\frac{1}{48}\right)
Multiplying or dividing an even number of negative terms equals a positive
48\times \frac{1}{48}
Reduce the numbers
1\times 1
Simplify
1
-\frac{1}{3}\left(x^{\prime}\right)^{2}+\frac{1}{3}\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}}{3}+\frac{1}{3}\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}}{3}+\frac{\left(y^{\prime}\right)^{2}}{3}=1
Solution
\frac{\left(y^{\prime}\right)^{2}}{3}-\frac{\left(x^{\prime}\right)^{2}}{3}=1
Show Solutions
