💡Keep us in your study toolbox - Bookmark us
bookmark us
close
girl-logo

Ask Questions

Home > Math Solver

Math Solver

Topics
Pre Algebra
Fractions
Decimals
Exponents
Radicals
Ratios
Integer
Percent
Algebra
Equations
Expressions
Inequalities
Functions
Sequences
Logarithmetic
Complex Numbers
Matrices
Pre Calculus
Equations
Inequalities
Functions
Calculus
Derivatives
Integrals
Limits
Series
Trigonometry
Equations
Inequalities
Functions
Simplify
Proof
Geometry
Statistics and Probability
Matrix
\frac{x^2}{36}+\frac{y^2}{9} = 1
Question
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Uh oh!

Ask Tutors

Identify the conic
\left(0,0\right)
Evaluate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Solution
\left(0,0\right)
Show Solutions Hide Solutions

Find the center of the ellipse

Find the foci of the ellipse

Find the vertices of the ellipse

Find the eccentricity of the ellipse

load more hide
Solve the equation
\begin{align}&x=2\sqrt{9-y^{2}}\\&x=-2\sqrt{9-y^{2}}\end{align}
Evaluate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Move the expression to the right-hand side and change its sign
\frac{x^{2}}{36}=1-\frac{y^{2}}{9}
Subtract the terms
More Steps Hide Steps
Evaluate
1-\frac{y^{2}}{9}
Reduce fractions to a common denominator
\frac{9}{9}-\frac{y^{2}}{9}
Add or subtract the terms
\frac{9-y^{2}}{9}
\frac{x^{2}}{36}=\frac{9-y^{2}}{9}
\text{Multiply both sides of the equation by }36
\frac{x^{2}}{36}\times 36=\frac{9-y^{2}}{9}\times 36
Multiply the terms
x^{2}=\frac{\left(9-y^{2}\right)\times 36}{9}
Evaluate
x^{2}=36-4y^{2}
Take the root of both sides of the equation and remember to use both positive and negative roots
x=\pm \sqrt{36-4y^{2}}
Simplify the expression
More Steps Hide Steps
Evaluate
\sqrt{36-4y^{2}}
Factor the expression
\sqrt{4\left(9-y^{2}\right)}
The root of a product is equal to the product of the roots of each factor
\sqrt{4}\times \sqrt{9-y^{2}}
Evaluate the root
More Steps Hide Steps
Evaluate
\sqrt{4}
\text{Write the number in exponential form with the base of }2
\sqrt{2^{2}}
\text{Reduce the index of the radical and exponent with }2
2
2\sqrt{9-y^{2}}
x=\pm 2\sqrt{9-y^{2}}
Solution
\begin{align}&x=2\sqrt{9-y^{2}}\\&x=-2\sqrt{9-y^{2}}\end{align}
Show Solutions Hide Solutions

\text{Solve for }x

\text{Solve for }y

Rewrite the equation
\begin{align}&r=\frac{6\sqrt{4-3\cos^{2}\left(\theta \right)}}{4-3\cos^{2}\left(\theta \right)}\\&r=-\frac{6\sqrt{4-3\cos^{2}\left(\theta \right)}}{4-3\cos^{2}\left(\theta \right)}\end{align}
Evaluate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Multiply both sides of the equation by LCD
\left(\frac{x^{2}}{36}+\frac{y^{2}}{9}\right)\times 36=1\times 36
Simplify the equation
More Steps Hide Steps
Evaluate
\left(\frac{x^{2}}{36}+\frac{y^{2}}{9}\right)\times 36
Apply the distributive property
\frac{x^{2}}{36}\times 36+\frac{y^{2}}{9}\times 36
Simplify
x^{2}+y^{2}\times 4
Simplify
x^{2}+4y^{2}
x^{2}+4y^{2}=1\times 36
Simplify the equation
x^{2}+4y^{2}=36
\text{To convert the equation to polar coordinates,substitute }x\text{ for }r\cos\left(\theta \right)\text{ and }y\text{ for }r\sin\left(\theta \right)
\left(\cos\left(\theta \right)\times r\right)^{2}+4\left(\sin\left(\theta \right)\times r\right)^{2}=36
Factor the expression
\left(\cos^{2}\left(\theta \right)+4\sin^{2}\left(\theta \right)\right)r^{2}=36
Simplify the expression
\left(1+3\sin^{2}\left(\theta \right)\right)r^{2}=36
Divide the terms
r^{2}=\frac{36}{1+3\sin^{2}\left(\theta \right)}
Evaluate the power
r=\pm \sqrt{\frac{36}{1+3\sin^{2}\left(\theta \right)}}
Simplify the expression
More Steps Hide Steps
Evaluate
\sqrt{\frac{36}{1+3\sin^{2}\left(\theta \right)}}
To take a root of a fraction,take the root of the numerator and denominator separately
\frac{\sqrt{36}}{\sqrt{1+3\sin^{2}\left(\theta \right)}}
Simplify the radical expression
More Steps Hide Steps
Evaluate
\sqrt{36}
\text{Write the number in exponential form with the base of }6
\sqrt{6^{2}}
\text{Reduce the index of the radical and exponent with }2
6
\frac{6}{\sqrt{1+3\sin^{2}\left(\theta \right)}}
Rewrite the expression
\frac{6}{\sqrt{4-3\cos^{2}\left(\theta \right)}}
Multiply by the Conjugate
\frac{6\sqrt{4-3\cos^{2}\left(\theta \right)}}{\sqrt{4-3\cos^{2}\left(\theta \right)}\times \sqrt{4-3\cos^{2}\left(\theta \right)}}
Calculate
\frac{6\sqrt{4-3\cos^{2}\left(\theta \right)}}{4-3\cos^{2}\left(\theta \right)}
r=\pm \frac{6\sqrt{4-3\cos^{2}\left(\theta \right)}}{4-3\cos^{2}\left(\theta \right)}
Solution
\begin{align}&r=\frac{6\sqrt{4-3\cos^{2}\left(\theta \right)}}{4-3\cos^{2}\left(\theta \right)}\\&r=-\frac{6\sqrt{4-3\cos^{2}\left(\theta \right)}}{4-3\cos^{2}\left(\theta \right)}\end{align}
Show Solutions Hide Solutions
Testing for symmetry
\textrm{Symmetry with respect to the origin}
Evaluate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
\text{To test if the graph of }\frac{x^{2}}{36}+\frac{y^{2}}{9}=1\text{ is symmetry with respect to the origin,substitute -x for x and -y for y}
\frac{\left(-x\right)^{2}}{36}+\frac{\left(-y\right)^{2}}{9}=1
Evaluate
More Steps Hide Steps
Evaluate
\frac{\left(-x\right)^{2}}{36}+\frac{\left(-y\right)^{2}}{9}
Evaluate the power
\frac{x^{2}}{36}+\frac{\left(-y\right)^{2}}{9}
Evaluate the power
\frac{x^{2}}{36}+\frac{y^{2}}{9}
Reduce fractions to a common denominator
\frac{x^{2}}{36}+\frac{y^{2}\times 4}{9\times 4}
Reorder the terms
\frac{x^{2}}{36}+\frac{4y^{2}}{36}
Add or subtract the terms
\frac{x^{2}+4y^{2}}{36}
\frac{x^{2}+4y^{2}}{36}=1
Solution
\textrm{Symmetry with respect to the origin}
Show Solutions Hide Solutions

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Find the first derivative
\frac{dy}{dx}=-\frac{x}{4y}
Calculate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Take the derivative of both sides
\frac{d}{dx}\left(\frac{x^{2}}{36}+\frac{y^{2}}{9}\right)=\frac{d}{dx}\left(1\right)
Calculate the derivative
More Steps Hide Steps
Evaluate
\frac{d}{dx}\left(\frac{x^{2}}{36}+\frac{y^{2}}{9}\right)
Use differentiation rules
\frac{d}{dx}\left(\frac{x^{2}}{36}\right)+\frac{d}{dx}\left(\frac{y^{2}}{9}\right)
Evaluate the derivative
More Steps Hide Steps
Evaluate
\frac{d}{dx}\left(\frac{x^{2}}{36}\right)
Rewrite the expression
\frac{\frac{d}{dx}\left(x^{2}\right)}{36}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
\frac{2x}{36}
Calculate
\frac{x}{18}
\frac{x}{18}+\frac{d}{dx}\left(\frac{y^{2}}{9}\right)
Evaluate the derivative
More Steps Hide Steps
Evaluate
\frac{d}{dx}\left(\frac{y^{2}}{9}\right)
Rewrite the expression
\frac{\frac{d}{dx}\left(y^{2}\right)}{9}
Evaluate the derivative
\frac{2y\frac{dy}{dx}}{9}
\frac{x}{18}+\frac{2y\frac{dy}{dx}}{9}
Calculate
\frac{x+4y\frac{dy}{dx}}{18}
\frac{x+4y\frac{dy}{dx}}{18}=\frac{d}{dx}\left(1\right)
Calculate the derivative
\frac{x+4y\frac{dy}{dx}}{18}=0
Cross multiply
x+4y\frac{dy}{dx}=18\times 0
Simplify the equation
x+4y\frac{dy}{dx}=0
Move the constant to the right side
4y\frac{dy}{dx}=0-x
Removing 0 doesn't change the value,so remove it from the expression
4y\frac{dy}{dx}=-x
\text{Multiply both sides of the equation by }\frac{1}{4y}
4y\frac{dy}{dx}\times \frac{1}{4y}=-x\times \frac{1}{4y}
To multiply the fractions,multiply the numerators and denominators separately
4y\frac{dy}{dx}\times \frac{1}{4y}=-\frac{x}{4y}
Solution
\frac{dy}{dx}=-\frac{x}{4y}
Show Solutions Hide Solutions

\text{Find the derivative with respect to }x

\text{Find the derivative with respect to }y

Find the second derivative
\frac{d^2y}{dx^2}=-\frac{4y^{2}+x^{2}}{16y^{3}}
Calculate
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1
Take the derivative of both sides
\frac{d}{dx}\left(\frac{x^{2}}{36}+\frac{y^{2}}{9}\right)=\frac{d}{dx}\left(1\right)
Calculate the derivative
More Steps Hide Steps
Evaluate
\frac{d}{dx}\left(\frac{x^{2}}{36}+\frac{y^{2}}{9}\right)
Use differentiation rules
\frac{d}{dx}\left(\frac{x^{2}}{36}\right)+\frac{d}{dx}\left(\frac{y^{2}}{9}\right)
Evaluate the derivative
More Steps Hide Steps
Evaluate
\frac{d}{dx}\left(\frac{x^{2}}{36}\right)
Rewrite the expression
\frac{\frac{d}{dx}\left(x^{2}\right)}{36}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
\frac{2x}{36}
Calculate
\frac{x}{18}
\frac{x}{18}+\frac{d}{dx}\left(\frac{y^{2}}{9}\right)
Evaluate the derivative
More Steps Hide Steps
Evaluate
\frac{d}{dx}\left(\frac{y^{2}}{9}\right)
Rewrite the expression
\frac{\frac{d}{dx}\left(y^{2}\right)}{9}
Evaluate the derivative
\frac{2y\frac{dy}{dx}}{9}
\frac{x}{18}+\frac{2y\frac{dy}{dx}}{9}
Calculate
\frac{x+4y\frac{dy}{dx}}{18}
\frac{x+4y\frac{dy}{dx}}{18}=\frac{d}{dx}\left(1\right)
Calculate the derivative
\frac{x+4y\frac{dy}{dx}}{18}=0
Cross multiply
x+4y\frac{dy}{dx}=18\times 0
Simplify the equation
x+4y\frac{dy}{dx}=0
Move the constant to the right side
4y\frac{dy}{dx}=0-x
Removing 0 doesn't change the value,so remove it from the expression
4y\frac{dy}{dx}=-x
\text{Multiply both sides of the equation by }\frac{1}{4y}
4y\frac{dy}{dx}\times \frac{1}{4y}=-x\times \frac{1}{4y}
To multiply the fractions,multiply the numerators and denominators separately
4y\frac{dy}{dx}\times \frac{1}{4y}=-\frac{x}{4y}
\text{Cancel out the greatest common factor }4y
\frac{dy}{dx}=-\frac{x}{4y}
Take the derivative of both sides
\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(-\frac{x}{4y}\right)
Calculate the derivative
\frac{d^2y}{dx^2}=\frac{d}{dx}\left(-\frac{x}{4y}\right)
Use differentiation rules
\frac{d^2y}{dx^2}=-\frac{\frac{d}{dx}\left(x\right)\times 4y-x\times \frac{d}{dx}\left(4y\right)}{\left(4y\right)^{2}}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
\frac{d^2y}{dx^2}=-\frac{1\times 4y-x\times \frac{d}{dx}\left(4y\right)}{\left(4y\right)^{2}}
Calculate the derivative
More Steps Hide Steps
Evaluate
\frac{d}{dx}\left(4y\right)
Use differentiation rules
\frac{d}{dy}\left(4y\right)\times \frac{dy}{dx}
Evaluate the derivative
4\frac{dy}{dx}
\frac{d^2y}{dx^2}=-\frac{1\times 4y-x\times 4\frac{dy}{dx}}{\left(4y\right)^{2}}
Any expression multiplied by 1 remains the same
\frac{d^2y}{dx^2}=-\frac{4y-x\times 4\frac{dy}{dx}}{\left(4y\right)^{2}}
Use the commutative property to reorder the terms
\frac{d^2y}{dx^2}=-\frac{4y-4x\frac{dy}{dx}}{\left(4y\right)^{2}}
Calculate
More Steps Hide Steps
Evaluate
\left(4y\right)^{2}
To raise a product to a power,raise each factor to that power
4^{2}y^{2}
Evaluate the power
16y^{2}
\frac{d^2y}{dx^2}=-\frac{4y-4x\frac{dy}{dx}}{16y^{2}}
Calculate
\frac{d^2y}{dx^2}=-\frac{y-x\frac{dy}{dx}}{4y^{2}}
\text{Use equation }\frac{dy}{dx}=-\frac{x}{4y}\text{ to substitute}
\frac{d^2y}{dx^2}=-\frac{y-x\left(-\frac{x}{4y}\right)}{4y^{2}}
Solution
More Steps Hide Steps
Calculate
-\frac{y-x\left(-\frac{x}{4y}\right)}{4y^{2}}
Multiply the terms
More Steps Hide Steps
Evaluate
x\left(-\frac{x}{4y}\right)
Multiplying or dividing an odd number of negative terms equals a negative
-x\times \frac{x}{4y}
Multiply the terms
-\frac{x\times x}{4y}
Multiply the terms
-\frac{x^{2}}{4y}
-\frac{y-\left(-\frac{x^{2}}{4y}\right)}{4y^{2}}
Subtract the terms
More Steps Hide Steps
Simplify
y-\left(-\frac{x^{2}}{4y}\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}}{4y}
Reduce fractions to a common denominator
\frac{y\times 4y}{4y}+\frac{x^{2}}{4y}
Reorder the terms
\frac{4y^{2}}{4y}+\frac{x^{2}}{4y}
Add or subtract the terms
\frac{4y^{2}+x^{2}}{4y}
-\frac{\frac{4y^{2}+x^{2}}{4y}}{4y^{2}}
Divide the terms
More Steps Hide Steps
Rewrite the expression
\frac{\frac{4y^{2}+x^{2}}{4y}}{4y^{2}}
Multiply by the reciprocal
\frac{4y^{2}+x^{2}}{4y}\times \frac{1}{4y^{2}}
Multiply the terms
\frac{4y^{2}+x^{2}}{4y\times 4y^{2}}
Multiply the terms
\frac{4y^{2}+x^{2}}{16y^{3}}
-\frac{4y^{2}+x^{2}}{16y^{3}}
\frac{d^2y}{dx^2}=-\frac{4y^{2}+x^{2}}{16y^{3}}
Show Solutions Hide Solutions

\text{Find the derivative with respect to }x

\text{Find the derivative with respect to }y

Graph

Still have questions?
Ask UpStudy online

  • 24/7 expert live tutors

  • Unlimited numbers of questions

  • Step-by-step explanations

Download UpStudy
📸 STUDY CAN BE A REAL STRUGGLE
Why Not UpStudy It?
Select your plan below
Premium

You can enjoy

  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to answer and
    solution
Basic
  • Limited Solutions