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Question
x^{2}+y^{2}=25
Identify the conic
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Find the radius of the circle
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Find the center of the circle
r=5
Rewrite in standard form
x^{2}+y^{2}=25
Solution
r=5
\left(0,0\right)
Rewrite in standard form
x^{2}+y^{2}=25
Solution
\left(0,0\right)
Show Solutions
Solve the equation
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\text{Solve for }x
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\text{Solve for }y
\begin{align}&x=\sqrt{25-y^{2}}\\&x=-\sqrt{25-y^{2}}\end{align}
Evaluate
x^{2}+y^{2}=25
Move the expression to the right-hand side and change its sign
x^{2}=25-y^{2}
Take the root of both sides of the equation and remember to use both positive and negative roots
x=\pm \sqrt{25-y^{2}}
Solution
\begin{align}&x=\sqrt{25-y^{2}}\\&x=-\sqrt{25-y^{2}}\end{align}
\begin{align}&y=\sqrt{25-x^{2}}\\&y=-\sqrt{25-x^{2}}\end{align}
Evaluate
x^{2}+y^{2}=25
Move the expression to the right-hand side and change its sign
y^{2}=25-x^{2}
Take the root of both sides of the equation and remember to use both positive and negative roots
y=\pm \sqrt{25-x^{2}}
Solution
\begin{align}&y=\sqrt{25-x^{2}}\\&y=-\sqrt{25-x^{2}}\end{align}
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
x^{2}+y^{2}=25
\text{To test if the graph of }x^{2}+y^{2}=25\text{ is symmetry with respect to the origin,substitute -x for x and -y for y}
\left(-x\right)^{2}+\left(-y\right)^{2}=25
Evaluate
More Steps
Evaluate
\left(-x\right)^{2}+\left(-y\right)^{2}
Rewrite the expression
x^{2}+\left(-y\right)^{2}
Rewrite the expression
x^{2}+y^{2}
x^{2}+y^{2}=25
Solution
\textrm{Symmetry with respect to the origin}
\textrm{Symmetry with respect to the x-axis}
Evaluate
x^{2}+y^{2}=25
\text{To test if the graph of }x^{2}+y^{2}=25\text{ is symmetry with respect to the x-axis,substitute -y for y}
x^{2}+\left(-y\right)^{2}=25
Evaluate
x^{2}+y^{2}=25
Solution
\textrm{Symmetry with respect to the x-axis}
\textrm{Symmetry with respect to the y-axis}
Evaluate
x^{2}+y^{2}=25
\text{To test if the graph of }x^{2}+y^{2}=25\text{ is symmetry with respect to the y-axis,substitute -x for x}
\left(-x\right)^{2}+y^{2}=25
Evaluate
x^{2}+y^{2}=25
Solution
\textrm{Symmetry with respect to the y-axis}
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{x}{y}
Calculate
x^{2}+y^{2}=25
Take the derivative of both sides
\frac{d}{dx}\left(x^{2}+y^{2}\right)=\frac{d}{dx}\left(25\right)
Calculate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(x^{2}+y^{2}\right)
Use differentiation rules
\frac{d}{dx}\left(x^{2}\right)+\frac{d}{dx}\left(y^{2}\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
2x+\frac{d}{dx}\left(y^{2}\right)
Evaluate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(y^{2}\right)
Use differentiation rules
\frac{d}{dy}\left(y^{2}\right)\times \frac{dy}{dx}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
2y\frac{dy}{dx}
2x+2y\frac{dy}{dx}
2x+2y\frac{dy}{dx}=\frac{d}{dx}\left(25\right)
Calculate the derivative
2x+2y\frac{dy}{dx}=0
Move the expression to the right-hand side and change its sign
2y\frac{dy}{dx}=0-2x
Removing 0 doesn't change the value,so remove it from the expression
2y\frac{dy}{dx}=-2x
Divide both sides
\frac{2y\frac{dy}{dx}}{2y}=\frac{-2x}{2y}
Divide the numbers
\frac{dy}{dx}=\frac{-2x}{2y}
Solution
More Steps
Evaluate
\frac{-2x}{2y}
\text{Cancel out the common factor }2
\frac{-x}{y}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
-\frac{x}{y}
\frac{dy}{dx}=-\frac{x}{y}
\frac{dx}{dy}=-\frac{y}{x}
Calculate
x^{2}+y^{2}=25
Take the derivative of both sides
\frac{d}{dy}\left(x^{2}+y^{2}\right)=\frac{d}{dy}\left(25\right)
Calculate the derivative
More Steps
Evaluate
\frac{d}{dy}\left(x^{2}+y^{2}\right)
Use differentiation rules
\frac{d}{dy}\left(x^{2}\right)+\frac{d}{dy}\left(y^{2}\right)
Evaluate the derivative
More Steps
Evaluate
\frac{d}{dy}\left(x^{2}\right)
Use differentiation rules
\frac{d}{dx}\left(x^{2}\right)\times \frac{dx}{dy}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
2x\frac{dx}{dy}
2x\frac{dx}{dy}+\frac{d}{dy}\left(y^{2}\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
2x\frac{dx}{dy}+2y
2x\frac{dx}{dy}+2y=\frac{d}{dy}\left(25\right)
Calculate the derivative
2x\frac{dx}{dy}+2y=0
Move the expression to the right-hand side and change its sign
2x\frac{dx}{dy}=0-2y
Removing 0 doesn't change the value,so remove it from the expression
2x\frac{dx}{dy}=-2y
Divide both sides
\frac{2x\frac{dx}{dy}}{2x}=\frac{-2y}{2x}
Divide the numbers
\frac{dx}{dy}=\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{dx}{dy}=-\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{y^{2}+x^{2}}{y^{3}}
Calculate
x^{2}+y^{2}=25
Take the derivative of both sides
\frac{d}{dx}\left(x^{2}+y^{2}\right)=\frac{d}{dx}\left(25\right)
Calculate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(x^{2}+y^{2}\right)
Use differentiation rules
\frac{d}{dx}\left(x^{2}\right)+\frac{d}{dx}\left(y^{2}\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
2x+\frac{d}{dx}\left(y^{2}\right)
Evaluate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(y^{2}\right)
Use differentiation rules
\frac{d}{dy}\left(y^{2}\right)\times \frac{dy}{dx}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
2y\frac{dy}{dx}
2x+2y\frac{dy}{dx}
2x+2y\frac{dy}{dx}=\frac{d}{dx}\left(25\right)
Calculate the derivative
2x+2y\frac{dy}{dx}=0
Move the expression to the right-hand side and change its sign
2y\frac{dy}{dx}=0-2x
Removing 0 doesn't change the value,so remove it from the expression
2y\frac{dy}{dx}=-2x
Divide both sides
\frac{2y\frac{dy}{dx}}{2y}=\frac{-2x}{2y}
Divide the numbers
\frac{dy}{dx}=\frac{-2x}{2y}
Divide the numbers
More Steps
Evaluate
\frac{-2x}{2y}
\text{Cancel out the common factor }2
\frac{-x}{y}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
-\frac{x}{y}
\frac{dy}{dx}=-\frac{x}{y}
Take the derivative of both sides
\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(-\frac{x}{y}\right)
Calculate the derivative
\frac{d^2y}{dx^2}=\frac{d}{dx}\left(-\frac{x}{y}\right)
Use differentiation rules
\frac{d^2y}{dx^2}=-\frac{\frac{d}{dx}\left(x\right)\times y-x\times \frac{d}{dx}\left(y\right)}{y^{2}}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
\frac{d^2y}{dx^2}=-\frac{1\times y-x\times \frac{d}{dx}\left(y\right)}{y^{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{1\times y-x\frac{dy}{dx}}{y^{2}}
Any expression multiplied by 1 remains the same
\frac{d^2y}{dx^2}=-\frac{y-x\frac{dy}{dx}}{y^{2}}
\text{Use equation }\frac{dy}{dx}=-\frac{x}{y}\text{ to substitute}
\frac{d^2y}{dx^2}=-\frac{y-x\left(-\frac{x}{y}\right)}{y^{2}}
Solution
More Steps
Calculate
-\frac{y-x\left(-\frac{x}{y}\right)}{y^{2}}
Multiply the terms
More Steps
Evaluate
x\left(-\frac{x}{y}\right)
Multiplying or dividing an odd number of negative terms equals a negative
-x\times \frac{x}{y}
Multiply the terms
-\frac{x\times x}{y}
Multiply the terms
-\frac{x^{2}}{y}
-\frac{y-\left(-\frac{x^{2}}{y}\right)}{y^{2}}
Subtract the terms
More Steps
Simplify
y-\left(-\frac{x^{2}}{y}\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
y+\frac{x^{2}}{y}
Reduce fractions to a common denominator
\frac{y\times y}{y}+\frac{x^{2}}{y}
Write all numerators above the common denominator
\frac{y\times y+x^{2}}{y}
Multiply the terms
\frac{y^{2}+x^{2}}{y}
-\frac{\frac{y^{2}+x^{2}}{y}}{y^{2}}
Divide the terms
More Steps
Evaluate
\frac{\frac{y^{2}+x^{2}}{y}}{y^{2}}
Multiply by the reciprocal
\frac{y^{2}+x^{2}}{y}\times \frac{1}{y^{2}}
Multiply the terms
\frac{y^{2}+x^{2}}{y\times y^{2}}
Multiply the terms
\frac{y^{2}+x^{2}}{y^{3}}
-\frac{y^{2}+x^{2}}{y^{3}}
\frac{d^2y}{dx^2}=-\frac{y^{2}+x^{2}}{y^{3}}
\frac{d^2x}{dy^2}=-\frac{x^{2}+y^{2}}{x^{3}}
Calculate
x^{2}+y^{2}=25
Take the derivative of both sides
\frac{d}{dy}\left(x^{2}+y^{2}\right)=\frac{d}{dy}\left(25\right)
Calculate the derivative
More Steps
Evaluate
\frac{d}{dy}\left(x^{2}+y^{2}\right)
Use differentiation rules
\frac{d}{dy}\left(x^{2}\right)+\frac{d}{dy}\left(y^{2}\right)
Evaluate the derivative
More Steps
Evaluate
\frac{d}{dy}\left(x^{2}\right)
Use differentiation rules
\frac{d}{dx}\left(x^{2}\right)\times \frac{dx}{dy}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
2x\frac{dx}{dy}
2x\frac{dx}{dy}+\frac{d}{dy}\left(y^{2}\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
2x\frac{dx}{dy}+2y
2x\frac{dx}{dy}+2y=\frac{d}{dy}\left(25\right)
Calculate the derivative
2x\frac{dx}{dy}+2y=0
Move the expression to the right-hand side and change its sign
2x\frac{dx}{dy}=0-2y
Removing 0 doesn't change the value,so remove it from the expression
2x\frac{dx}{dy}=-2y
Divide both sides
\frac{2x\frac{dx}{dy}}{2x}=\frac{-2y}{2x}
Divide the numbers
\frac{dx}{dy}=\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{dx}{dy}=-\frac{y}{x}
Take the derivative of both sides
\frac{d}{dy}\left(\frac{dx}{dy}\right)=\frac{d}{dy}\left(-\frac{y}{x}\right)
Calculate the derivative
\frac{d^2x}{dy^2}=\frac{d}{dy}\left(-\frac{y}{x}\right)
Use differentiation rules
\frac{d^2x}{dy^2}=-\frac{\frac{d}{dy}\left(y\right)\times x-y\times \frac{d}{dy}\left(x\right)}{x^{2}}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
\frac{d^2x}{dy^2}=-\frac{1\times x-y\times \frac{d}{dy}\left(x\right)}{x^{2}}
Calculate the derivative
More Steps
Evaluate
\frac{d}{dy}\left(x\right)
Use differentiation rules
\frac{d}{dx}\left(x\right)\times \frac{dx}{dy}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
\frac{dx}{dy}
\frac{d^2x}{dy^2}=-\frac{1\times x-y\frac{dx}{dy}}{x^{2}}
Any expression multiplied by 1 remains the same
\frac{d^2x}{dy^2}=-\frac{x-y\frac{dx}{dy}}{x^{2}}
\text{Use equation }\frac{dx}{dy}=-\frac{y}{x}\text{ to substitute}
\frac{d^2x}{dy^2}=-\frac{x-y\left(-\frac{y}{x}\right)}{x^{2}}
Solution
More Steps
Calculate
-\frac{x-y\left(-\frac{y}{x}\right)}{x^{2}}
Multiply the terms
More Steps
Evaluate
y\left(-\frac{y}{x}\right)
Multiplying or dividing an odd number of negative terms equals a negative
-y\times \frac{y}{x}
Multiply the terms
-\frac{y\times y}{x}
Multiply the terms
-\frac{y^{2}}{x}
-\frac{x-\left(-\frac{y^{2}}{x}\right)}{x^{2}}
Subtract the terms
More Steps
Simplify
x-\left(-\frac{y^{2}}{x}\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
x+\frac{y^{2}}{x}
Reduce fractions to a common denominator
\frac{x\times x}{x}+\frac{y^{2}}{x}
Write all numerators above the common denominator
\frac{x\times x+y^{2}}{x}
Multiply the terms
\frac{x^{2}+y^{2}}{x}
-\frac{\frac{x^{2}+y^{2}}{x}}{x^{2}}
Divide the terms
More Steps
Evaluate
\frac{\frac{x^{2}+y^{2}}{x}}{x^{2}}
Multiply by the reciprocal
\frac{x^{2}+y^{2}}{x}\times \frac{1}{x^{2}}
Multiply the terms
\frac{x^{2}+y^{2}}{x\times x^{2}}
Multiply the terms
\frac{x^{2}+y^{2}}{x^{3}}
-\frac{x^{2}+y^{2}}{x^{3}}
\frac{d^2x}{dy^2}=-\frac{x^{2}+y^{2}}{x^{3}}
Show Solutions
Rewrite the equation
\begin{align}&r=5\\&r=-5\end{align}
Evaluate
x^{2}+y^{2}=25
\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
\left(\cos\left(\theta \right)\times r\right)^{2}+\left(\sin\left(\theta \right)\times r\right)^{2}=25
Factor the expression
\left(\cos^{2}\left(\theta \right)+\sin^{2}\left(\theta \right)\right)r^{2}=25
Simplify the expression
r^{2}=25
Evaluate the power
r=\pm \sqrt{25}
Evaluate the square root
More Steps
Evaluate
\sqrt{25}
\text{Write the number in exponential form with the base of }5
\sqrt{5^{2}}
\text{Reduce the index of the radical and exponent with }2
5
r=\pm 5
Solution
\begin{align}&r=5\\&r=-5\end{align}
Show Solutions
Graph
