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Question
2xy=15
Solve the equation
-
\text{Solve for }x
-
\text{Solve for }y
x=\frac{15}{2y}
Evaluate
2xy=15
Rewrite the expression
2yx=15
Divide both sides
\frac{2yx}{2y}=\frac{15}{2y}
Solution
x=\frac{15}{2y}
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
2xy=15
\text{To test if the graph of }2xy=15\text{ is symmetry with respect to the origin,substitute -x for x and -y for y}
2\left(-x\right)\left(-y\right)=15
Evaluate
2xy=15
Solution
\textrm{Symmetry with respect to the origin}
Show Solutions
Rewrite the equation
\begin{align}&r=\frac{\sqrt{15\sin\left(2\theta \right)}}{\left|\sin\left(2\theta \right)\right|}\\&r=-\frac{\sqrt{15\sin\left(2\theta \right)}}{\left|\sin\left(2\theta \right)\right|}\end{align}
Evaluate
2xy=15
\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
2\cos\left(\theta \right)\times r\sin\left(\theta \right)\times r=15
Factor the expression
2\cos\left(\theta \right)\sin\left(\theta \right)\times r^{2}=15
Simplify the expression
\sin\left(2\theta \right)\times r^{2}=15
Divide the terms
r^{2}=\frac{15}{\sin\left(2\theta \right)}
Evaluate the power
r=\pm \sqrt{\frac{15}{\sin\left(2\theta \right)}}
Simplify the expression
More Steps
Evaluate
\sqrt{\frac{15}{\sin\left(2\theta \right)}}
To take a root of a fraction,take the root of the numerator and denominator separately
\frac{\sqrt{15}}{\sqrt{\sin\left(2\theta \right)}}
Multiply by the Conjugate
\frac{\sqrt{15}\times \sqrt{\sin\left(2\theta \right)}}{\sqrt{\sin\left(2\theta \right)}\times \sqrt{\sin\left(2\theta \right)}}
Calculate
\frac{\sqrt{15}\times \sqrt{\sin\left(2\theta \right)}}{\left|\sin\left(2\theta \right)\right|}
The product of roots with the same index is equal to the root of the product
\frac{\sqrt{15\sin\left(2\theta \right)}}{\left|\sin\left(2\theta \right)\right|}
r=\pm \frac{\sqrt{15\sin\left(2\theta \right)}}{\left|\sin\left(2\theta \right)\right|}
Solution
\begin{align}&r=\frac{\sqrt{15\sin\left(2\theta \right)}}{\left|\sin\left(2\theta \right)\right|}\\&r=-\frac{\sqrt{15\sin\left(2\theta \right)}}{\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
2xy=15
Take the derivative of both sides
\frac{d}{dx}\left(2xy\right)=\frac{d}{dx}\left(15\right)
Calculate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(2xy\right)
Use differentiation rules
\frac{d}{dx}\left(2x\right)\times y+2x\times \frac{d}{dx}\left(y\right)
Evaluate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(2x\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
2\times \frac{d}{dx}\left(x\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
2\times 1
Any expression multiplied by 1 remains the same
2
2y+2x\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}
2y+2x\frac{dy}{dx}
2y+2x\frac{dy}{dx}=\frac{d}{dx}\left(15\right)
Calculate the derivative
2y+2x\frac{dy}{dx}=0
Move the expression to the right-hand side and change its sign
2x\frac{dy}{dx}=0-2y
Removing 0 doesn't change the value,so remove it from the expression
2x\frac{dy}{dx}=-2y
Divide both sides
\frac{2x\frac{dy}{dx}}{2x}=\frac{-2y}{2x}
Divide the numbers
\frac{dy}{dx}=\frac{-2y}{2x}
Solution
More Steps
Evaluate
\frac{-2y}{2x}
\text{Cancel out the common factor }2
\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
2xy=15
Take the derivative of both sides
\frac{d}{dx}\left(2xy\right)=\frac{d}{dx}\left(15\right)
Calculate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(2xy\right)
Use differentiation rules
\frac{d}{dx}\left(2x\right)\times y+2x\times \frac{d}{dx}\left(y\right)
Evaluate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(2x\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
2\times \frac{d}{dx}\left(x\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
2\times 1
Any expression multiplied by 1 remains the same
2
2y+2x\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}
2y+2x\frac{dy}{dx}
2y+2x\frac{dy}{dx}=\frac{d}{dx}\left(15\right)
Calculate the derivative
2y+2x\frac{dy}{dx}=0
Move the expression to the right-hand side and change its sign
2x\frac{dy}{dx}=0-2y
Removing 0 doesn't change the value,so remove it from the expression
2x\frac{dy}{dx}=-2y
Divide both sides
\frac{2x\frac{dy}{dx}}{2x}=\frac{-2y}{2x}
Divide the numbers
\frac{dy}{dx}=\frac{-2y}{2x}
Divide the numbers
More Steps
Evaluate
\frac{-2y}{2x}
\text{Cancel out the common factor }2
\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}}{15}-\frac{\left(y^{\prime}\right)^{2}}{15}=1
Evaluate
2xy=15
Move the expression to the left side
2xy-15=0
\text{The coefficients A,B and C of the general equation are A=}0\text{,B=}2\text{ and C=}0
\begin{align}&A=0\\&B=2\\&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}{2}
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 }2xy-15=0
2\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)-15=0
Calculate
More Steps
Calculate
2\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)-15
Use the commutative property to reorder the terms
2\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)-15
Use the commutative property to reorder the terms
2\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)-15
Use the commutative property to reorder the terms
2\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)-15
Use the commutative property to reorder the terms
2\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)-15
Expand the expression
More Steps
Calculate
2\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(\sqrt{2}\times x^{\prime}-\sqrt{2}\times y^{\prime}\right)\left(\frac{\sqrt{2}}{2}x^{\prime}+\frac{\sqrt{2}}{2}y^{\prime}\right)
Apply the distributive property
\sqrt{2}\times x^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}+\sqrt{2}\times x^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}-\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}-\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the terms
\left(x^{\prime}\right)^{2}+\sqrt{2}\times x^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}-\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}-\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the numbers
\left(x^{\prime}\right)^{2}+x^{\prime}y^{\prime}-\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}x^{\prime}-\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the numbers
\left(x^{\prime}\right)^{2}+x^{\prime}y^{\prime}-y^{\prime}x^{\prime}-\sqrt{2}\times y^{\prime}\times \frac{\sqrt{2}}{2}y^{\prime}
Multiply the terms
\left(x^{\prime}\right)^{2}+x^{\prime}y^{\prime}-y^{\prime}x^{\prime}-\left(y^{\prime}\right)^{2}
Subtract the terms
\left(x^{\prime}\right)^{2}+0-\left(y^{\prime}\right)^{2}
Removing 0 doesn't change the value,so remove it from the expression
\left(x^{\prime}\right)^{2}-\left(y^{\prime}\right)^{2}
\left(x^{\prime}\right)^{2}-\left(y^{\prime}\right)^{2}-15
\left(x^{\prime}\right)^{2}-\left(y^{\prime}\right)^{2}-15=0
Move the constant to the right-hand side and change its sign
\left(x^{\prime}\right)^{2}-\left(y^{\prime}\right)^{2}=0-\left(-15\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
\left(x^{\prime}\right)^{2}-\left(y^{\prime}\right)^{2}=0+15
Removing 0 doesn't change the value,so remove it from the expression
\left(x^{\prime}\right)^{2}-\left(y^{\prime}\right)^{2}=15
\text{Multiply both sides of the equation by }\frac{1}{15}
\left(\left(x^{\prime}\right)^{2}-\left(y^{\prime}\right)^{2}\right)\times \frac{1}{15}=15\times \frac{1}{15}
Multiply the terms
More Steps
Evaluate
\left(\left(x^{\prime}\right)^{2}-\left(y^{\prime}\right)^{2}\right)\times \frac{1}{15}
Use the the distributive property to expand the expression
\left(x^{\prime}\right)^{2}\times \frac{1}{15}-\left(y^{\prime}\right)^{2}\times \frac{1}{15}
Use the commutative property to reorder the terms
\frac{1}{15}\left(x^{\prime}\right)^{2}-\left(y^{\prime}\right)^{2}\times \frac{1}{15}
Use the commutative property to reorder the terms
\frac{1}{15}\left(x^{\prime}\right)^{2}-\frac{1}{15}\left(y^{\prime}\right)^{2}
\frac{1}{15}\left(x^{\prime}\right)^{2}-\frac{1}{15}\left(y^{\prime}\right)^{2}=15\times \frac{1}{15}
Multiply the terms
More Steps
Evaluate
15\times \frac{1}{15}
Reduce the numbers
1\times 1
Simplify
1
\frac{1}{15}\left(x^{\prime}\right)^{2}-\frac{1}{15}\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}}{15}-\frac{1}{15}\left(y^{\prime}\right)^{2}=1
Solution
\frac{\left(x^{\prime}\right)^{2}}{15}-\frac{\left(y^{\prime}\right)^{2}}{15}=1
Show Solutions
