You CAN Ace Calculus

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

An ellipse is formed when a plane crosses that is not parallel to one of sides of the cone and not parallel to the axis of the cone. A circle is a special form of an ellipse where the plane is perpendicular to the axis of the cone.
On this page, we discuss ellipses in rectangular coordinates. For a discussion of ellipses in polar form, see this separate page.

 Ellipse

The standard equation for an ellipse is $$\displaystyle{ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 }$$. Figure 2 contains more information than we need right now but it will suffice. The longer axis is called the major axis (in this plot it is horizontal). The shorter axis is called the minor axis. The vertices are located on the ellipse where it crosses the major axis. The foci are also on the major axis, labeled F1 and F2 on this plot.

The major axis is determined by the denominators, $$a^2$$ and $$b^2$$. The larger value is in the denominator of the major axis, i.e. if $$a > b$$ then the major axis is parallel to the x-axis. We need to define a value c where $$c^2=\abs{a^2-b^2}$$ which will help us determine the location of the foci.

These tables contain the main attributes of an ellipse. We assume here that $$a > b$$. Similar equations exist for $$a < b$$.

$$\displaystyle{ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 }$$

center

$$(h,k)$$

major axis

$$y=k$$

vertices

$$(h \pm a, k),$$ $$(h, k \pm b)$$

foci

$$(h \pm c,k)$$

$$c^2=\abs{a^2-b^2}$$

eccentricity

$$e=c/a$$

$$\displaystyle{ \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 }$$

center

$$(h,k)$$

major axis

$$x=h$$

vertices

$$(h, k \pm a),$$ $$(h \pm b, k)$$

foci

$$(h,k \pm c)$$

$$c^2=\abs{a^2-b^2}$$

eccentricity

$$e=c/a$$

 Notes

1. Since the foci are closer to the center than the vertices, it follows that $$c < a$$ and therefore $$0 < e < 1$$.
2. Notice in the standard form of the equation, both terms are positive. This is how you know the graph is an ellipse and not a hyperbola.
3. In the general form of the equation, $$Ax^2+Bxy+Cy^2+$$ $$Dx+Ey+F=0$$, $$A > 0$$ and $$C > 0$$.
4. The eccentricity e is not the same as the irrational constant $$e \approx 2.72$$.

Okay, time for some fun videos about ellipses. Here are a couple of videos about playing pool on an elliptical table. They clearly show the relationship between the foci and demonstrate some fun physics at the same time.

### Numberphile - Elliptical Pool Table (1) [3min-39secs]

video by Numberphile

### Numberphile - Elliptical Pool Table (2) [5min-52secs]

video by Numberphile

### Practice

Write the standard form of the equation for an ellipse, centered at the origin, vertical major axis of length 8 and minor axis of length 2.

Problem Statement

Write the standard form of the equation for an ellipse, centered at the origin, vertical major axis of length 8 and minor axis of length 2.

Solution

### 1587 solution video

video by PatrickJMT

Write the standard form of the equation for an ellipse, centered at the origin, with x-intercepts at $$\pm 12$$ and foci at $$(0,\pm 5)$$.

Problem Statement

Write the standard form of the equation for an ellipse, centered at the origin, with x-intercepts at $$\pm 12$$ and foci at $$(0,\pm 5)$$.

Solution

### 1588 solution video

video by PatrickJMT

Write the standard form of the equation for an ellipse, centered at the origin, with minor axis of length 6 and foci at $$(\pm 8, 0)$$.

Problem Statement

Write the standard form of the equation for an ellipse, centered at the origin, with minor axis of length 6 and foci at $$(\pm 8, 0)$$.

Solution

### 1589 solution video

video by PatrickJMT

Find the intercepts of the ellipse $$\displaystyle{ \frac{y^2}{100} + \frac{x^2}{121} = 1 }$$.

Problem Statement

Find the intercepts of the ellipse $$\displaystyle{ \frac{y^2}{100} + \frac{x^2}{121} = 1 }$$.

Solution

### 1590 solution video

video by PatrickJMT

Graph the ellipse $$\displaystyle{ 1 - \frac{y^2}{16} = x^2 }$$.

Problem Statement

Graph the ellipse $$\displaystyle{ 1 - \frac{y^2}{16} = x^2 }$$.

Solution

### 1591 solution video

video by PatrickJMT

Find the center and the radius of the circle $$x^2+2x+y^2=4$$.

Problem Statement

Find the center and the radius of the circle $$x^2+2x+y^2=4$$.

Solution

### 1594 solution video

video by Krista King Math

Sketch the circle $$x^2 + y^2 = 4x$$.

Problem Statement

Sketch the circle $$x^2 + y^2 = 4x$$.

Solution

### 1595 solution video

video by Krista King Math

Sketch the circle $$x^2 + y^2 + 6y = 0$$.

Problem Statement

Sketch the circle $$x^2 + y^2 + 6y = 0$$.

Solution

### 1596 solution video

video by Krista King Math

Sketch the circle $$x^2 + y^2 + 2x + 2y = 2$$.

Problem Statement

Sketch the circle $$x^2 + y^2 + 2x + 2y = 2$$.

Solution

### 1597 solution video

video by Krista King Math

Sketch the circle $$x^2 + y^2 + 10x - 20y + 100 = 0$$.

Problem Statement

Sketch the circle $$x^2 + y^2 + 10x - 20y + 100 = 0$$.

Solution

### 1598 solution video

video by Krista King Math

Sketch the circle $$2x^2 + 2y^2 + 2x - 2y = 1$$.

Problem Statement

Sketch the circle $$2x^2 + 2y^2 + 2x - 2y = 1$$.

Solution

### 1599 solution video

video by Krista King Math

Sketch the circle $$9x^2 + 9y^2 - 6x - 12y = 11$$.

Problem Statement

Sketch the circle $$9x^2 + 9y^2 - 6x - 12y = 11$$.

Solution

### 1600 solution video

video by Krista King Math

Graph $$\displaystyle{ \frac{x^2}{9} + \frac{y^2}{5} = 1 }$$.

Problem Statement

Graph $$\displaystyle{ \frac{x^2}{9} + \frac{y^2}{5} = 1 }$$.

Solution

### 1601 solution video

video by PatrickJMT