20.4: Ellipses

Definitions and Theorems

Definition: Ellipse

An ellipse is the set of all points in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).

Definition: Major and Minor Axes, Vertex and Co-Vertex, and Center of an Ellipse

The longer axis of an ellipse is called the major axis, and the shorter axis is called the minor axis. Each endpoint of the major axis is called a vertex of the ellipse (plural: vertices), and each endpoint of the minor axis is called a co-vertex of the ellipse. The center of the ellipse is the midpoint of both the major and minor axes.

Theorem: Standard Form of the Equation of an Ellipse Centered at the Origin

The standard form of the equation of an ellipse with center \((0,0)\) and major axis on the \(x\)-axis is\[\dfrac{x^2}{v_x^2}+\dfrac{y^2}{v_y^2}=1, \nonumber \]where

• \( v_x \) and \( v_y \) are positive
• \(v_x > v_y\)
• the length of the major axis is \(2 v_x\)
• the coordinates of the vertices are \(( \pm v_x, 0)\)
• the length of the minor axis is \(2 v_y\)
• the coordinates of the co-vertices are \((0, \pm v_y)\)
• the coordinates of the foci are \(( \pm c, 0)\), where \(c^2= \left| v_x^2 - v_y^2 \right| \).

The standard form of the equation of an ellipse with center \((0,0)\) and major axis on the \(y\)-axis is\[ \dfrac{x^2}{v_x^2}+\dfrac{y^2}{v_y^2}=1, \nonumber \]where

• \( v_x \) and \( v_y \) are positive
• \(v_y > v_x \)
• the length of the major axis is \(2 v_y\)
• the coordinates of the vertices are \((0, \pm v_y)\)
• the length of the minor axis is \(2 v_x\)
• the coordinates of the co-vertices are \(( \pm v_x, 0)\)
• the coordinates of the foci are \((0, \pm c)\), where \(c^2 = \left| v_x^2 - v_y^2 \right| \).

Proof

To derive the equation of an ellipse centered at the origin, we begin with the foci \( (−c,0) \) and \( (c,0) \). The ellipse is the set of all points \( (x,y) \) such that the sum of the distances from \( (x,y) \) to the foci is constant, as shown in the figure below.

If \((v_x, 0)\) is a vertex of the ellipse (on the major axis), the distance from \((-c, 0)\) to \((v_x, 0)\) is \(v_x-(-c)=v_x+c\). The distance from \((c, 0)\) to \((v_x, 0)\) is \(v_x-c\). The sum of the distances from the foci to the vertex is\[(v_x+c)+(v_x-c) = 2v_x \nonumber \]If \((x, y)\) is a point on the ellipse, then we can define the following variables:\[ \begin{array}{rcl} d_1 & = & \text { the distance from }(-c, 0) \text { to }(x, y) \\[6pt] d_2 & = & \text { the distance from }(c, 0) \text { to }(x, y)\\[6pt] \end{array} \nonumber \]By the definition of an ellipse, \(d_1+d_2\) is constant for any point \((x, y)\) on the ellipse. We know that the sum of these distances is \(2 v_x\) for the vertex \((v_x, 0)\). It follows that \(d_1+d_2=2 v_x\) for any point on the ellipse. We will begin the derivation by applying the Distance Formula. The rest of the derivation is algebraic.\[ \begin{array}{rrclcl}
& d_1 + d_2 & = & 2v_x & & \\[6pt]
\implies & \sqrt{\left( x - (-c) \right)^2 + \left( y - 0 \right)^2} + \sqrt{\left( x - c \right)^2 + \left( y - 0 \right)^2} & = & 2v_x & \quad & \left( \text{substituting} \right) \\[6pt]
\implies & \sqrt{\left( x + c \right)^2 + y^2} + \sqrt{\left( x - c \right)^2 + y^2} & = & 2v_x & \quad & \left( \text{simplifying} \right) \\[6pt]
\implies & \sqrt{\left( x + c \right)^2 + y^2} & = & 2v_x - \sqrt{\left( x - c \right)^2 + y^2} & \quad & \left( \text{isolating a radical} \right) \\[6pt]
\implies & \left( x + c \right)^2 + y^2 & = & \left[ 2v_x - \sqrt{\left( x - c \right)^2 + y^2} \right]^2 & \quad & \left( \text{squaring both sides} \right) \\[6pt]
\implies & \left( x + c \right)^2 + y^2 & = & 4v_x^2 - 4v_x \sqrt{\left( x - c \right)^2 + y^2} + \left( x - c \right)^2 + y^2 & \quad & \left( \text{distributing} \right) \\[6pt]
\implies & x^2 + 2 c x + c^2 + y^2 & = & 4v_x^2 - 4v_x \sqrt{\left( x - c \right)^2 + y^2} + x^2 - 2c x + c^2 + y^2 & \quad & \left( \text{distributing} \right) \\[6pt]
\implies & 2 c x & = & 4v_x^2 - 4v_x \sqrt{\left( x - c \right)^2 + y^2} - 2c x & \quad & \left( \text{subtracting }x^2, \, c^2, \text{ and }y^2 \right. \\[6pt]
 &  &  & & \quad & \left. \text{ from both sides} \right) \\[6pt]
\implies & 4 c x - 4v_x^2 & = & -4v_x \sqrt{\left( x - c \right)^2 + y^2} & \quad & \left( \text{isolating the radical} \right) \\[6pt]
\implies & c x - v_x^2 & = & -v_x \sqrt{\left( x - c \right)^2 + y^2} & \quad & \left( \text{dividing both sides by }4 \right) \\[6pt]
\implies & \left(c x - v_x^2\right)^2 & = & v_x^2 \left(\left( x - c \right)^2 + y^2\right) & \quad & \left( \text{squaring both sides} \right) \\[6pt]
\implies & c^2 x^2 - 2v_x^2 c x + v_x^4 & = & v_x^2 \left(x^2 - 2cx + c^2 + y^2\right) & \quad & \left( \text{distributing} \right) \\[6pt]
\implies & c^2 x^2 - 2v_x^2 c x + v_x^4 & = & v_x^2 x^2 - 2 v_x^2 cx + v_x^2 c^2 + v_x^2 y^2 & \quad & \left( \text{distributing} \right) \\[6pt]
\implies & c^2 x^2 + v_x^4 & = & v_x^2 x^2 + v_x^2 c^2 + v_x^2 y^2 & \quad & \left( \text{adding }2v_x^2 c x\text{ to both sides} \right) \\[6pt]
\implies & v_x^4 - v_x^2 c^2 & = & v_x^2 x^2 - c^2 x^2 + v_x^2 y^2 & \quad & \left( \text{subtracting }v_x^2 c^2 \text{ and }c^2 x^2 \right. \\[6pt]
 &  &  & & \quad & \left. \text{ from both sides} \right) \\[6pt]
\implies & v_x^2(v_x^2 - c^2) & = & x^2(v_x^2 - c^2 ) + v_x^2 y^2 & \quad & \left( \text{factor} \right) \\[6pt]
\implies & v_x^2 b^2 & = & x^2 b^2 + v_x^2 y^2 & \quad & \left( \text{let }b^2 = v_x^2 - c^2 \right) \\[6pt]
\implies & \dfrac{v_x^2 b^2}{v_x^2 b^2} & = & \dfrac{x^2 b^2}{v_x^2 b^2} + \dfrac{v_x^2 y^2}{v_x^2 b^2} & \quad & \left( \text{divide both sides by }v_x^2 b^2 \right) \\[6pt]
\implies & 1 & = & \dfrac{x^2}{v_x^2} + \dfrac{y^2}{b^2} & \quad & \left( \text{simplify} \right) \\[6pt]
\end{array} \nonumber \]If we let \( x = 0 \) in our final result, we get \( y^2 = b^2 \), which implies \( y = \pm b \). Therefore, the point \( \left( 0,b \right) \) must be the co-vertex of the ellipse. As such, we relabel \( b = v_y \) to arrive at the standard equation of an ellipse:\[\dfrac{x^2}{v_x^2}+\dfrac{y^2}{v_y^2}=1. \nonumber \]This equation defines an ellipse centered at the origin. If \( v_x > v_y\), the ellipse is stretched further in the horizontal direction, and if \(v_y>v_x\), the ellipse is stretched further in the vertical direction.

Theorem: Standard Form of the Equation of an Ellipse Centered Off-Origin

The standard form of the equation of an ellipse with center \((h, k)\) and major axis parallel to the \(x\)-axis is\[\dfrac{(x-h)^2}{v_x^2}+\dfrac{(y-k)^2}{v_y^2}=1, \nonumber \]where

• \( v_x \) and \( v_y \) are positive
• \(v_x^2 > v_y^2 \)
• the length of the major axis is \(2 v_x\)
• the coordinates of the vertices are \((h \pm v_x, k)\)
• the length of the minor axis is \(2 v_y\)
• the coordinates of the co-vertices are \((h, k \pm v_y)\)
• the coordinates of the foci are \((h \pm c, k)\), where \(c^2 = \left| v_x^2 - v_y^2 \right| \).

The standard form of the equation of an ellipse with center \((h, k)\) and major axis parallel to the \(y\)-axis is\[\dfrac{(x-h)^2}{v_x^2}+\dfrac{(y-k)^2}{v_y^2}=1, \nonumber \]where

• \( v_x \) and \( v_y \) are positive
• \(v_y^2 > v_x^2 \)
• the length of the major axis is \(2 v_y\)
• the coordinates of the vertices are \((h, k \pm v_y)\)
• the length of the minor axis is \(2 v_x\)
• the coordinates of the co-vertices are \((h \pm v_x, k)\)
• the coordinates of the foci are \((h, k \pm c)\), where \(c^2 = \left| v_x^2 - v_y^2 \right| \).