# Step By Step Calculus » 3.3 - Ellipse

Synopsis

Given two fixed points in the plane, the curve traced out by a point which moves so that the sum of its distances from two fixed points is always a constant is called an ellipse. These two fixed points are called the foci of the ellipse. An ellipse with its two foci at (\pm p,0) (\pm p,0) is expressed by the equation

If both foci coincide, i.e. p=0 p=0 then a=b a=b which means the ellipse becomes a circle.

\displaystyle {\textbf{(Equation of Ellipse)}\qquad\qquad \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \;\;\textrm{ where }b^2=a^2-p^2.}\displaystyle {\textbf{(Equation of Ellipse)}\qquad\qquad \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \;\;\textrm{ where }b^2=a^2-p^2.} |

The Standard Form of a general ellipse is:

where (h,k) (h,k) is the center of the ellipse and p^2=a^2-b^2 p^2=a^2-b^2.

\displaystyle {\textbf{(Standard Form) }\qquad\left\{\begin{array}{ll}
\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 & \textrm{ horizontal Major, foci }(h-p,k),(h+p,k)\\
\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1 & \textrm{ vertical Major, foci }(h,k-p),(h,k+p)\end{array}\right.,}\displaystyle {\textbf{(Standard Form) }\qquad\left\{\begin{array}{ll}
\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 & \textrm{ horizontal Major, foci }(h-p,k),(h+p,k)\\
\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1 & \textrm{ vertical Major, foci }(h,k-p),(h,k+p)\end{array}\right.,} |

We prefer to use the following STAPLED procedure when sketching an ellipse.

Standard Convert the given equation to standard form.

Type Indicate whether vertical or horizontal major axis.

Axes Draw a pair of xy xy-axes on the xy xy-plane. Use the same scale for both.

Points Mark the foci on the xy xy-plane.

Lines Draw the Major and Minor axes line segments.

Ends Make the ends of the Major and Minor axes clear.

Draw Draw the ellipse: Make sure it goes through the ends of the Major and Minor Axes. Also, ensure that the sum of the distances from each point on the ellipse to the two foci is constant.