Ellipses

An ellipse has two foci (plural of focus) and two directrices (plural of directrix). We’ll examine the directrices in detail later in this section.

Figure 8 shows a typical ellipse with its essential components labeled. Understanding these parts will help you work with ellipse equations and solve related problems. Let’s examine each component and its coordinates in this standard position.

• The Foci: Points [latex]F[/latex] and [latex]F'[/latex] are both the same distance [latex]c[/latex] from the origin. Their coordinates are [latex]F(c,0)[/latex] and [latex]F'(-c,0)[/latex].
• The Major Axis: The longest distance across the ellipse. Points [latex]P[/latex] and [latex]P'[/latex] mark the ends of the major axis with coordinates [latex]\left(a,0\right)[/latex] and  [latex]\left(-a,0\right)[/latex]. The major axis length is [latex]2a[/latex].
• The Vertices: Points [latex]P[/latex] and [latex]P'[/latex] are called the vertices of the ellipse—the endpoints of the major axis.
• The Minor Axis: The shortest distance across the ellipse, perpendicular to the major axis. Points [latex]Q[/latex] and [latex]Q'[/latex] mark its ends with coordinates [latex]\left(0,b\right)[/latex] and [latex]\left(0,-b\right)[/latex].

Now we’ll use these components to derive the standard form equation of an ellipse. According to the ellipse definition, the sum of distances from any point on the ellipse to the two foci remains constant. Let’s use point [latex]P[/latex] to find this constant sum.

Since [latex]P[/latex] has coordinates [latex]\left(a,0\right)[/latex], the sum of the distances is

This means for any arbitrary point [latex]A(x,y)[/latex] on the ellipse, the sum also equals [latex]2a[/latex]. Using the distance formula:

To solve for the ellipse equation, we’ll eliminate the radicals through careful algebra. Subtract the second radical from both sides and square:

Isolate the radical and square again:

Rearrange the variables on the left-hand side of the equation and the constants on the right-hand side:

Divide both sides by [latex]{a}^{2}-{c}^{2}[/latex]. This gives the equation:

When the center moves from the origin to point [latex]\left(h,k\right)[/latex], we get standard form of an ellipse.

An ellipse is called horizontal when its major axis is horizontal, and vertical when its major axis is vertical.

The general form of an ellipse is [latex]Ax^2 + By^2 + Cx + Dy + E = 0[/latex], where [latex]A[/latex] and [latex]B[/latex] are both positive or both negative. To convert from general to standard form, use completing the square.

Put the equation [latex]9{x}^{2}+4{y}^{2}-36x+24y+36=0[/latex] into standard form and graph the resulting ellipse.

Show Solution

First subtract 36 from both sides of the equation:

[latex]9{x}^{2}+4{y}^{2}-36x+24y=-36[/latex].

 

Next group the x terms together and the y terms together, and factor out the common factor:

[latex]\begin{array}{ccc}\hfill \left(9{x}^{2}-36x\right)+\left(4{y}^{2}+24y\right)& =\hfill & -36\hfill \\ \hfill 9\left({x}^{2}-4x\right)+4\left({y}^{2}+6y\right)& =\hfill & -36.\hfill \end{array}[/latex]

 

We need to determine the constant that, when added inside each set of parentheses, results in a perfect square. In the first set of parentheses, take half the coefficient of x and square it. This gives [latex]{\left(\frac{-4}{2}\right)}^{2}=4[/latex]. In the second set of parentheses, take half the coefficient of y and square it. This gives [latex]{\left(\frac{6}{2}\right)}^{2}=9[/latex]. Add these inside each pair of parentheses. Since the first set of parentheses has a 9 in front, we are actually adding 36 to the left-hand side. Similarly, we are adding 36 to the second set as well. Therefore the equation becomes

[latex]\begin{array}{}\\ 9\left({x}^{2}-4x+4\right)+4\left({y}^{2}+6y+9\right)=-36+36+36\hfill \\ 9\left({x}^{2}-4x+4\right)+4\left({y}^{2}+6y+9\right)=36.\hfill \end{array}[/latex]

 

Now factor both sets of parentheses and divide by 36:

[latex]\begin{array}{}\\ \hfill 9{\left(x - 2\right)}^{2}+4{\left(y+3\right)}^{2}& =\hfill & 36\hfill \\ \hfill \frac{9{\left(x - 2\right)}^{2}}{36}+\frac{4{\left(y+3\right)}^{2}}{36}& =\hfill & 1\hfill \\ \hfill \frac{{\left(x - 2\right)}^{2}}{4}+\frac{{\left(y+3\right)}^{2}}{9}& =\hfill & 1.\hfill \end{array}[/latex]

 

The equation is now in standard form. Comparing this to the theorem equation gives [latex]h=2[/latex], [latex]k=-3[/latex], [latex]a=3[/latex], and [latex]b=2[/latex]. This is a vertical ellipse with center at [latex]\left(2,-3\right)[/latex], major axis 6, and minor axis 4. The graph of this ellipse appears as follows.

Ellipses appear throughout nature and human-made structures, creating fascinating phenomena that affect everything from planetary motion to architectural acoustics.

Planetary Orbits: According to Kepler’s first law of planetary motion, planets orbit the Sun in elliptical paths with the Sun at one focus (Figure 11a). This means Earth’s distance from the Sun varies throughout the year. Other celestial objects follow elliptical orbits too. Comets like Halley’s Comet, moons orbiting planets, and satellites orbiting Earth all travel in elliptical paths.

Whispering Galleries: Ellipses have a remarkable reflective property: a light ray from one focus reflects off the ellipse and passes through the other focus. Sound waves behave the same way, creating “whispering galleries.” The National Statuary Hall in the U.S. Capitol demonstrates this acoustic property perfectly. This elliptical room served as the House of Representatives meeting place for nearly fifty years. Floor marks identify the two foci, and people standing on these spots can hear each other clearly across the room, even when it’s crowded.