Section 10.1The Ellipse

Corequisite Skills review (optional warm-up)

Learning Objectives

• Complete the square of a binomial expression. (IA 9.2.1)
• Graph a circle. (IA 11.1.4)

Objective 1: Complete the square of a binomial expression. (IA 9.2.1)

Vocabulary

Fill in the blanks.

We say that \({x}^{2}+4x+4\) is a ________ square trinomial because \({x}^{2}+4x+4={( + )}^{2}\) .

The square root property states that \({(x+2)}^{2}=9\) means that \((x+2)=\) ________ and \((x+2)=-\) ________.

Complete the square of a binomial expression.

Try It #1

Use the square root property to solve \({(x+3)}^{2}=5\) . Use the steps below to guide your work.

• ⓐ Take a square root of both sides. Remember to have ± on the right side.
• ⓑ Solve for x.

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Complete the square of a binomial expression.

Try It #2

Use the square root property to solve \({x}^{2}+6x+9=7\) . Use the steps below to guide your work.

• ⓐ \({x}^{2}+6x+9\) is a perfect square trinomial. Factor it into \({(\_+\_)}^{2}\) .
• ⓑ Take a square root of both sides. Remember to have ± on the right side.
• ⓒ Solve for x.

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But what happens if we have to solve an equation where the trinomial is not a perfect square?

For example, \({x}^{2}+4x+5=2\) ? For these types of equations, we can use a process called completing the square.

Recall \({(x+1)}^{2}=(x+1)(x+1)={x}^{2}+x+x+1={x}^{2}+2+1\) .

We can use the Binomial Squares Pattern to make a perfect square.

Warm-up Example 1

Complete the square for \({x}^{2}+6x\) to make it a perfect square.

Show solution

Table 1

Since there is a plus sign between the two terms, we will use the (a + b)2 pattern \({a}^{2}+2ab+{b}^{2}={(a+b)}^{2}\) .	\({x}^{2}+6x\)
We ultimately need to find the last term of this trinomial that will make it a perfect square trinomial. To do that we will need to find b. But first we start with determining a. Notice that the first term of x2 + 6x is a square, x2. This tells us that a = x.	\({x}^{2}+2·x·b+{b}^{2}\)
What number, b, when multiplied with 2x, gives 6x? It would have to be 3, which is (½)(6). So b = 3.	\({x}^{2}+2·3·x+\_\)
Now to complete the perfect square trinomial, we will find the last term by squaring b, which is 32 = 9.	\({x}^{2}+6x+9\)
We can now factor.	\({(x+3)}^{2}\)

So, we found that adding 9 to x² + 6x completes the square, and we write it as (x + 3)².

How To

• Identify b, the coefficient of x.
• Find \(\frac{1}{2}{b}^{2}\) , the number needed to complete the square.
• Add \(\frac{1}{2}{b}^{2}\) to \({x}^{2}+bx\)
• Factor the perfect square trinomial, writing it as a binomial squared.

Practice Makes Perfect

Determine what number would have to be added to the given terms to create a perfect square trinomial. Then rewrite as a binomial squared.

P1

\({x}^{2}+12x\)

P2

\({x}^{2}+5x\)

P3

\({x}^{2}-\frac{1}{2}x\)

P4

\({x}^{2}+\frac{3}{2}x\)

Objective 2: Graph a circle. (IA 11.1.4)

A circle is all points in a plane that are a fixed distance from a given point in the plane. The given point is called the center, (h, k) and the fixed distance is called the radius, r, of the circle.

The standard or graphing form of the equation of a circle with center, (h, k) and radius, r, is \({(x-h)}^{2}+{(y-k)}^{2}={r}^{2}\) .

Warm-up Example 2

Write the standard (graphing) form of the equation of the circle with radius 2 and center (-1, 3).

Show solution

Table 2

Use the standard, (graphing) form of the equation of a circle.	\({(x-h)}^{2}+{(y-k)}^{2}={r}^{2}\)
Substitute in the values (h, k)=(1, -3), where h=1, k=-3.	\((x-{(-1))}^{2}+(y-{(3))}^{2}={2}^{2}\)
Simplify.	\({(x+1)}^{2}+{(y-3)}^{2}=4\)

Warm-up Example 3Graph a circle

Find the center and radius, then graph the circle: \({(x+2)}^{2}+{(y-1)}^{2}=9\)

Show solution

Table 3

Use the standard (graphing) form of the equation of a circle.	\({(x-h)}^{2}+{(y-k)}^{2}={r}^{2}\)
Identify the center, (h,k) and radius, r.	\((x-{(-2))}^{2}+{(y-1)}^{2}={3}^{2}\)
Graph the circle.	Center (-2, 1), r=3

The general form of the equation of a circle is \({x}^{2}+{y}^{2}+ax+by+c=0\) . If we are given an equation in general form, we can change it to standard, also called the graphing form, by completing the squares in both x and y. Then we can graph the circle using its center and radius.

Warm-up Example 4

Find the center and radius, then graph the circle: \({x}^{2}+{y}^{2}-4x-6y+4=0\)

Show solution

We need to rewrite this general form into standard (graphing) form in order to find the center and radius.

Table 4

Group the x-terms and y-terms. Collect the constants on the right side.	\({x}^{2}-4x+{y}^{2}-6y=-4\)
Complete the squares.	\({x}^{2}-4x+4+{y}^{2}-6y+9=-4+4+9\)
Rewrite as binomial squares.	\({(x-2)}^{2}+{(y-3)}^{2}=9\)
Identify the center and radius.	Center (2, 3), r=3
Graph the circle.

Practice Makes Perfect

P5

Write the standard (graphing) form of the equation of the circle with radius 4 and center (2,–5).

P6

Find the center and radius, then graph the circle: \({(x-3)}^{2}+{(y+1)}^{2}=4\)

P7

Find the center and radius, then graph the circle: \({x}^{2}+{y}^{2}-6x-8y+9=0\)

Can you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at the other end? The National Statuary Hall in Washington, D.C., shown in Figure 1, is such a room. (Architect of the Capitol. http://www.aoc.gov. Accessed April 15, 2014.) It is an semi-circular room called a whispering chamber because the shape makes it possible for sound to travel along the walls and dome. In this section, we will investigate the shape of this room and its real-world applications, including how far apart two people in Statuary Hall can stand and still hear each other whisper.

Writing Equations of Ellipses in Standard Form

A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape, as shown in Figure 2.

Conic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the variable terms determine the shape. This section focuses on the four variations of the standard form of the equation for the ellipse. An ellipse is the set of all points \((x,y)\) 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).

We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. The result is an ellipse. See Figure 3.

Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis. Each endpoint of the major axis is the vertex of the ellipse (plural: vertices), and each endpoint of the minor axis is a co-vertex of the ellipse. The center of an ellipse is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci. See Figure 4.

In this section, we restrict ellipses to those that are positioned vertically or horizontally in the coordinate plane. That is, the axes will either lie on or be parallel to the x- and y-axes. Later in the chapter, we will see ellipses that are rotated in the coordinate plane.

To work with horizontal and vertical ellipses in the coordinate plane, we consider two cases: those that are centered at the origin and those that are centered at a point other than the origin. First we will learn to derive the equations of ellipses, and then we will learn how to write the equations of ellipses in standard form. Later we will use what we learn to draw the graphs.

Deriving the Equation of an Ellipse Centered at the Origin

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 Figure 5.

If \((a,0)\) is a vertex of the ellipse, the distance from \((-c,0)\) to \((a,0)\) is \(a-(-c)=a+c.\) The distance from \((c,0)\) to \((a,0)\) is \(a-c\) . The sum of the distances from the foci to the vertex is

\[(a+c)+(a-c)=2a\]

If \((x,y)\) is a point on the ellipse, then we can define the following variables:

\[\begin{array}{l}{d}_{1}=\text{the distance from }(-c,0)\,\text{to }(x,y) \\ {d}_{2}=\text{the distance from }(c,0)\,\text{to }(x,y)\end{array}\]

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 \(2a\) for the vertex \((a,0).\) It follows that \({d}_{1}+{d}_{2}=2a\) for any point on the ellipse. We will begin the derivation by applying the distance formula. The rest of the derivation is algebraic.

\[\begin{array}{ll}{d}_{1}+{d}_{2}=\sqrt{{(x-(-c))}^{2}+{(y-0)}^{2}}+\sqrt{{(x-c)}^{2}+{(y-0)}^{2}}=2a & \text{Distance formula} \\ \sqrt{{(x+c)}^{2}+{y}^{2}}+\sqrt{{(x-c)}^{2}+{y}^{2}}=2a & \text{Simplify expressions}\text{.} \\ \sqrt{{(x+c)}^{2}+{y}^{2}}=2a-\sqrt{{(x-c)}^{2}+{y}^{2}} & \text{Move radical to opposite side}\text{.} \\ {(x+c)}^{2}+{y}^{2}={[2a-\sqrt{{(x-c)}^{2}+{y}^{2}}]}^{2} & \text{Square both sides}\text{.} \\ {x}^{2}+2cx+{c}^{2}+{y}^{2}=4{a}^{2}-4a\sqrt{{(x-c)}^{2}+{y}^{2}}+{(x-c)}^{2}+{y}^{2} & \text{Expand the squares}\text{.} \\ {x}^{2}+2cx+{c}^{2}+{y}^{2}=4{a}^{2}-4a\sqrt{{(x-c)}^{2}+{y}^{2}}+{x}^{2}-2cx+{c}^{2}+{y}^{2} & \text{Expand remaining squares}\text{.} \\ 2cx=4{a}^{2}-4a\sqrt{{(x-c)}^{2}+{y}^{2}}-2cx & \text{Combine like terms}\text{.} \\ 4cx-4{a}^{2}=-4a\sqrt{{(x-c)}^{2}+{y}^{2}} & \text{Isolate the radical}\text{.} \\ cx-{a}^{2}=-a\sqrt{{(x-c)}^{2}+{y}^{2}} & \text{Divide by 4}\text{.} \\ {[cx-{a}^{2}]}^{2}={a}^{2}{[\sqrt{{(x-c)}^{2}+{y}^{2}}]}^{2} & \text{Square both sides}\text{.} \\ {c}^{2}{x}^{2}-2{a}^{2}cx+{a}^{4}={a}^{2}({x}^{2}-2cx+{c}^{2}+{y}^{2}) & \text{Expand the squares}\text{.} \\ {c}^{2}{x}^{2}-2{a}^{2}cx+{a}^{4}={a}^{2}{x}^{2}-2{a}^{2}cx+{a}^{2}{c}^{2}+{a}^{2}{y}^{2} & \text{Distribute }{a}^{2}. \\ {a}^{2}{x}^{2}-{c}^{2}{x}^{2}+{a}^{2}{y}^{2}={a}^{4}-{a}^{2}{c}^{2} & \text{Rewrite}\text{.} \\ {x}^{2}({a}^{2}-{c}^{2})+{a}^{2}{y}^{2}={a}^{2}({a}^{2}-{c}^{2}) & \text{Factor common terms}\text{.} \\ {x}^{2}{b}^{2}+{a}^{2}{y}^{2}={a}^{2}{b}^{2} & \text{Set }{b}^{2}={a}^{2}-{c}^{2}. \\ \frac{{x}^{2}{b}^{2}}{{a}^{2}{b}^{2}}+\frac{{a}^{2}{y}^{2}}{{a}^{2}{b}^{2}}=\frac{{a}^{2}{b}^{2}}{{a}^{2}{b}^{2}} & \text{Divide both sides by }{a}^{2}{b}^{2}. \\ \frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1 & \text{Simplify}\text{.}\end{array}\]

Thus, the standard equation of an ellipse is \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1.\) This equation defines an ellipse centered at the origin. If \(a>b,\) the ellipse is stretched further in the horizontal direction, and if \(b>a,\) the ellipse is stretched further in the vertical direction.

Writing Equations of Ellipses Centered at the Origin in Standard Form

Standard forms of equations tell us about key features of graphs. Take a moment to recall some of the standard forms of equations we’ve worked with in the past: linear, quadratic, cubic, exponential, logarithmic, and so on. By learning to interpret standard forms of equations, we are bridging the relationship between algebraic and geometric representations of mathematical phenomena.

The key features of the ellipse are its center, vertices, co-vertices, foci, and lengths and positions of the major and minor axes. Just as with other equations, we can identify all of these features just by looking at the standard form of the equation. There are four variations of the standard form of the ellipse. These variations are categorized first by the location of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form a mental picture of the ellipse.

Standard Forms of the Equation of an Ellipse with Center (0,0)

The standard form of the equation of an ellipse with center \((0,0)\) and major axis on the x-axis is

\[\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\]

where

• \(a>b\)
• the length of the major axis is \(2a\)
• the coordinates of the vertices are \((\pm a,0)\)
• the length of the minor axis is \(2b\)
• the coordinates of the co-vertices are \((0,\pm b)\)
• the coordinates of the foci are \((\pm c,0)\) , where \({c}^{2}={a}^{2}-{b}^{2}.\) See Figure 6a

The standard form of the equation of an ellipse with center \((0,0)\) and major axis on the y-axis is

\[\frac{{x}^{2}}{{b}^{2}}+\frac{{y}^{2}}{{a}^{2}}=1\]

where

• \(a>b\)
• the length of the major axis is \(2a\)
• the coordinates of the vertices are \((0,\pm a)\)
• the length of the minor axis is \(2b\)
• the coordinates of the co-vertices are \((\pm b,0)\)
• the coordinates of the foci are \((0,\pm c)\) , where \({c}^{2}={a}^{2}-{b}^{2}.\) See Figure 6b

Note that the vertices, co-vertices, and foci are related by the equation \({c}^{2}={a}^{2}-{b}^{2}.\) When we are given the coordinates of the foci and vertices of an ellipse, we can use this relationship to find the equation of the ellipse in standard form.

Writing Equations of Ellipses Not Centered at the Origin

Like the graphs of other equations, the graph of an ellipse can be translated. If an ellipse is translated \(h\) units horizontally and \(k\) units vertically, the center of the ellipse will be \((h,k).\) This translation results in the standard form of the equation we saw previously, with \(x\) replaced by \((x-h)\) and y replaced by \((y-k).\)

Standard Forms of the Equation of an Ellipse with Center (h, k)

The standard form of the equation of an ellipse with center \((h,\,\,k)\) and major axis parallel to the x-axis is

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

where

• \(a>b\)
• the length of the major axis is \(2a\)
• the coordinates of the vertices are \((h\pm a,k)\)
• the length of the minor axis is \(2b\)
• the coordinates of the co-vertices are \((h,k\pm b)\)
• the coordinates of the foci are \((h\pm c,k),\) where \({c}^{2}={a}^{2}-{b}^{2}.\) See Figure 7a

The standard form of the equation of an ellipse with center \((h,k)\) and major axis parallel to the y-axis is

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

where

• \(a>b\)
• the length of the major axis is \(2a\)
• the coordinates of the vertices are \((h,k\pm a)\)
• the length of the minor axis is \(2b\)
• the coordinates of the co-vertices are \((h\pm b,k)\)
• the coordinates of the foci are \((h,k\pm c),\) where \({c}^{2}={a}^{2}-{b}^{2}.\) See Figure 7b

Just as with ellipses centered at the origin, ellipses that are centered at a point \((h,k)\) have vertices, co-vertices, and foci that are related by the equation \({c}^{2}={a}^{2}-{b}^{2}.\) We can use this relationship along with the midpoint and distance formulas to find the equation of the ellipse in standard form when the vertices and foci are given.

Graphing Ellipses Centered at the Origin

Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard form \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,\,\,a>b\) for horizontal ellipses and \(\frac{{x}^{2}}{{b}^{2}}+\frac{{y}^{2}}{{a}^{2}}=1,\,\,a>b\) for vertical ellipses.

Example 3

Graph the ellipse given by the equation, \(\frac{{x}^{2}}{9}+\frac{{y}^{2}}{25}=1.\) Identify and label the center, vertices, co-vertices, and foci.

Read a and b as the square roots of the denominators, then find c with \({c}^{2}={a}^{2}-{b}^{2}\) to locate the foci.

Show solution

First, we determine the position of the major axis. Because \(25>9,\) the major axis is on the y-axis. Therefore, the equation is in the form \(\frac{{x}^{2}}{{b}^{2}}+\frac{{y}^{2}}{{a}^{2}}=1,\) where \({b}^{2}=9\) and \({a}^{2}=25.\) It follows that:

• the center of the ellipse is \((0,0)\)
• the coordinates of the vertices are \((0,\pm a)=(0,\pm \sqrt{25})=(0,\pm 5)\)
• the coordinates of the co-vertices are \((\pm b,0)=(\pm \sqrt{9},0)=(\pm 3,0)\)
• the coordinates of the foci are \((0,\pm c),\) where \({c}^{2}={a}^{2}-{b}^{2}\) Solving for \(c,\) we have:

\[\begin{array}{l}c=\pm \sqrt{{a}^{2}-{b}^{2}} \\ =\pm \sqrt{25-9} \\ =\pm \sqrt{16} \\ =\pm 4\end{array}\]

Therefore, the coordinates of the foci are \((0,\pm 4).\)

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse. See Figure 8.

Try It #5

Graph the ellipse given by the equation \(\frac{{x}^{2}}{36}+\frac{{y}^{2}}{4}=1.\) Identify and label the center, vertices, co-vertices, and foci.

Show answer

center: \((0,0);\) vertices: \((\pm 6,0);\) co-vertices: \((0,\pm 2);\) foci: \((\pm 4\sqrt{2},0)\)

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Example 4

Graph the ellipse given by the equation \(4{x}^{2}+25{y}^{2}=100.\) Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.

Divide every term by 100 first to get the equation into standard form, then read off a, b, and c.

Show solution

First, use algebra to rewrite the equation in standard form.

\[\begin{array}{l} 4{x}^{2}+25{y}^{2}=100 \\ \,\frac{4{x}^{2}}{100}+\frac{25{y}^{2}}{100}=\frac{100}{100} \\ \,\frac{{x}^{2}}{25}+\frac{{y}^{2}}{4}=1\end{array}\]

Next, we determine the position of the major axis. Because \(25>4,\) the major axis is on the x-axis. Therefore, the equation is in the form \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,\) where \({a}^{2}=25\) and \({b}^{2}=4.\) It follows that:

• the center of the ellipse is \((0,0)\)
• the coordinates of the vertices are \((\pm a,0)=(\pm \sqrt{25},0)=(\pm 5,0)\)
• the coordinates of the co-vertices are \((0,\pm b)=(0,\pm \sqrt{4})=(0,\pm 2)\)
• the coordinates of the foci are \((\pm c,0),\) where \({c}^{2}={a}^{2}-{b}^{2}.\) Solving for \(c,\) we have:

\[\begin{array}{l}c=\pm \sqrt{{a}^{2}-{b}^{2}} \\ =\pm \sqrt{25-4} \\ =\pm \sqrt{21}\end{array}\]

Therefore the coordinates of the foci are \((\pm \sqrt{21},0).\)

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.

Try It #6

Graph the ellipse given by the equation \(49{x}^{2}+16{y}^{2}=784.\) Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.

Show answer

Standard form: \(\frac{{x}^{2}}{16}+\frac{{y}^{2}}{49}=1;\) center: \((0,0);\) vertices: \((0,\pm 7);\) co-vertices: \((\pm 4,0);\) foci: \((0,\pm \sqrt{33})\)

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Graphing Ellipses Not Centered at the Origin

When an ellipse is not centered at the origin, we can still use the standard forms to find the key features of the graph. When the ellipse is centered at some point, \((h,k),\) we use the standard forms \(\frac{{(x-h)}^{2}}{{a}^{2}}+\frac{{(y-k)}^{2}}{{b}^{2}}=1,\,\,a>b\) for horizontal ellipses and \(\frac{{(x-h)}^{2}}{{b}^{2}}+\frac{{(y-k)}^{2}}{{a}^{2}}=1,\,\,a>b\) for vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.

Example 5

Graph the ellipse given by the equation, \(\frac{{(x+2)}^{2}}{4}+\frac{{(y-5)}^{2}}{9}=1.\) Identify and label the center, vertices, co-vertices, and foci.

Identify the center (h,k) from the equation first, then read a and b from the denominators to plot vertices and co-vertices around that center.

Show solution

First, we determine the position of the major axis. Because \(9>4,\) the major axis is parallel to the y-axis. Therefore, the equation is in the form \(\frac{{(x-h)}^{2}}{{b}^{2}}+\frac{{(y-k)}^{2}}{{a}^{2}}=1,\) where \({b}^{2}=4\) and \({a}^{2}=9.\) It follows that:

• the center of the ellipse is \((h,k)=(-2,\text{5})\)
• the coordinates of the vertices are \((h,k\pm a)=(-2,5\pm \sqrt{9})=(-2,5\pm 3),\) or \((-2,\text{2})\) and \((-2,\text{8})\)
• the coordinates of the co-vertices are \((h\pm b,k)=(-2\pm \sqrt{4},5)=(-2\pm 2,5),\) or \((-4,5)\) and \((0,\text{5})\)
• the coordinates of the foci are \((h,k\pm c),\) where \({c}^{2}={a}^{2}-{b}^{2}.\) Solving for \(c,\) we have:

\[\begin{array}{l}\begin{array}{l} \\ c=\pm \sqrt{{a}^{2}-{b}^{2}}\end{array} \\ =\pm \sqrt{9-4} \\ =\pm \sqrt{5}\end{array}\]

Therefore, the coordinates of the foci are \((-2,\text{5}-\sqrt{5})\) and \((-2,\text{5+}\sqrt{5}).\)

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.

Try It #7

Graph the ellipse given by the equation \(\frac{{(x-4)}^{2}}{36}+\frac{{(y-2)}^{2}}{20}=1.\) Identify and label the center, vertices, co-vertices, and foci.

Show answer

Center: \((4,2);\) vertices: \((-2,2)\) and \((10,2);\) co-vertices: \((4,2-2\sqrt{5})\) and \((4,2+2\sqrt{5});\) foci: \((0,2)\) and \((8,2)\)

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Example 6

Graph the ellipse given by the equation \(4{x}^{2}+9{y}^{2}-40x+36y+100=0.\) Identify and label the center, vertices, co-vertices, and foci.

Complete the square in both x and y to rewrite the equation in standard form, then identify the center, a, and b.

Show solution

We must begin by rewriting the equation in standard form.

\[4{x}^{2}+9{y}^{2}-40x+36y+100=0\]

Group terms that contain the same variable, and move the constant to the opposite side of the equation.

\[(4{x}^{2}-40x)+(9{y}^{2}+36y)=-100\]

Factor out the coefficients of the squared terms.

\[4({x}^{2}-10x)+9({y}^{2}+4y)=-100\]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

\[4({x}^{2}-10x+25)+9({y}^{2}+4y+4)=-100+100+36\]

Rewrite as perfect squares.

\[4{(x-5)}^{2}+9{(y+2)}^{2}=36\]

Divide both sides by the constant term to place the equation in standard form.

\[\frac{{(x-5)}^{2}}{9}+\frac{{(y+2)}^{2}}{4}=1\]

Now that the equation is in standard form, we can determine the position of the major axis. Because \(9>4,\) the major axis is parallel to the x-axis. Therefore, the equation is in the form \(\frac{{(x-h)}^{2}}{{a}^{2}}+\frac{{(y-k)}^{2}}{{b}^{2}}=1,\) where \({a}^{2}=9\) and \({b}^{2}=4.\) It follows that:

• the center of the ellipse is \((h,k)=(5,-2)\)
• the coordinates of the vertices are \((h\pm a,k)=(5\pm \sqrt{9},-2)=(5\pm 3,-2),\) or \((2,-2)\) and \((8,-2)\)
• the coordinates of the co-vertices are \((h,k\pm b)=(\text{5},-2\pm \sqrt{4})=(\text{5},-2\pm 2),\) or \((5,-4)\) and \((5,\text{0})\)
• the coordinates of the foci are \((h\pm c,k),\) where \({c}^{2}={a}^{2}-{b}^{2}.\) Solving for \(c,\) we have:

\[\begin{array}{l}c=\pm \sqrt{{a}^{2}-{b}^{2}} \\ =\pm \sqrt{9-4} \\ =\pm \sqrt{5}\end{array}\]

Therefore, the coordinates of the foci are \((\text{5}-\sqrt{5},-2)\) and \((\text{5+}\sqrt{5},-2).\)

Next we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse as shown in Figure 11.

Solving Applied Problems Involving Ellipses

Many real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves. Some buildings, called whispering chambers, are designed with elliptical domes so that a person whispering at one focus can easily be heard by someone standing at the other focus. This occurs because of the acoustic properties of an ellipse. When a sound wave originates at one focus of a whispering chamber, the sound wave will be reflected off the elliptical dome and back to the other focus. See Figure 12. In the whisper chamber at the Museum of Science and Industry in Chicago, two people standing at the foci—about 43 feet apart—can hear each other whisper. When these chambers are placed in unexpected places, such as the ones inside Bush International Airport in Houston and Grand Central Terminal in New York City, they can induce surprised reactions among travelers.

Example 7

A large room in an art gallery is a whispering chamber. Its dimensions are 46 feet wide by 96 feet long as shown in Figure 13.

• What is the standard form of the equation of the ellipse representing the outline of the room? Hint: assume a horizontal ellipse, and let the center of the room be the point \((0,0).\)
• If two visitors standing at the foci of this room can hear each other whisper, how far apart are the two visitors? Round to the nearest foot.

Model the room as an ellipse using half the width and length as b and a, then find c to locate the foci where the whisperers should stand.

Show solution

• We are assuming a horizontal ellipse with center \((0,0),\) so we need to find an equation of the form \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,\) where \(a>b.\) We know that the length of the major axis, \(2a,\) is longer than the length of the minor axis, \(2b.\) So the length of the room, 96, is represented by the major axis, and the width of the room, 46, is represented by the minor axis. Therefore, the equation of the ellipse is \(\frac{{x}^{2}}{2304}+\frac{{y}^{2}}{529}=1.\)
  • Solving for \(a,\) we have \(2a=96,\) so \(a=48,\) and \({a}^{2}=2304.\)
  • Solving for \(b,\) we have \(2b=46,\) so \(b=23,\) and \({b}^{2}=529.\)

  Therefore, the equation of the ellipse is \(\frac{{x}^{2}}{2304}+\frac{{y}^{2}}{529}=1.\)

• To find the distance between the senators, we must find the distance between the foci, \((\pm c,0),\) where \({c}^{2}={a}^{2}-{b}^{2}.\) Solving for \(c,\) we have: The points \((\pm 42,0)\) represent the foci. Thus, the distance between the senators is \(2(42)=84\) feet.

  \[\begin{array}{ll}{c}^{2}={a}^{2}-{b}^{2} & \\ {c}^{2}=2304-529 & \begin{array}{llll} & & & \end{array}\text{Substitute using the values found in part (a)}. \\ c=\pm \sqrt{2304-529} & \begin{array}{llll} & & & \end{array}\text{Take the square root of both sides}. \\ c=\pm \sqrt{1775} & \begin{array}{llll} & & & \end{array}\text{Subtract}. \\ c\approx \pm 42 & \begin{array}{llll} & & & \end{array}\text{Round to the nearest foot}.\end{array}\]

  The points \((\pm 42,0)\) represent the foci. Thus, the distance between the senators is \(2(42)=84\) feet.

Key Equations

Key Concepts

• An ellipse is the set of all points \((x,y)\) 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).
• When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. See Example 1 and Example 2.
• When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. See Example 3 and Example 4.
• When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. See Example 5 and Example 6.
• Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. See Example 7.

Section Exercises

Verbal

1

Define an ellipse in terms of its foci.

Show answer

An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.

2

Where must the foci of an ellipse lie?

3

What special case of the ellipse do we have when the major and minor axis are of the same length?

Show answer

This special case would be a circle.

4

For the special case mentioned in the previous question, what would be true about the foci of that ellipse?

5

What can be said about the symmetry of the graph of an ellipse with center at the origin and foci along the y-axis?

Show answer

It is symmetric about the x-axis, y-axis, and the origin.

Algebraic

For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.

6

\(2{x}^{2}+y=4\)

7

\(4{x}^{2}+9{y}^{2}=36\)

Show answer

yes; \(\frac{{x}^{2}}{{3}^{2}}+\frac{{y}^{2}}{{2}^{2}}=1\)

8

\(4{x}^{2}-{y}^{2}=4\)

9

\(4{x}^{2}+9{y}^{2}=1\)

Show answer

yes; \(\frac{{x}^{2}}{{(\frac{1}{2})}^{2}}+\frac{{y}^{2}}{{(\frac{1}{3})}^{2}}=1\)

10

\(4{x}^{2}-8x+9{y}^{2}-72y+112=0\)

For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.

11

\(\frac{{x}^{2}}{4}+\frac{{y}^{2}}{49}=1\)

Show answer

\(\frac{{x}^{2}}{{2}^{2}}+\frac{{y}^{2}}{{7}^{2}}=1;\) Endpoints of major axis \((0,7)\) and \((0,-7).\) Endpoints of minor axis \((2,0)\) and \((-2,0).\) Foci at \((0,3\sqrt{5}),(0,-3\sqrt{5}).\)

12

\(\frac{{x}^{2}}{100}+\frac{{y}^{2}}{64}=1\)

13

\({x}^{2}+9{y}^{2}=1\)

Show answer

\(\frac{{x}^{2}}{{(1)}^{2}}+\frac{{y}^{2}}{{(\frac{1}{3})}^{2}}=1;\) Endpoints of major axis \((1,0)\) and \((-1,0).\) Endpoints of minor axis \((0,\frac{1}{3}),(0,-\frac{1}{3}).\) Foci at \((\frac{2\sqrt{2}}{3},0),(-\frac{2\sqrt{2}}{3},0).\)

14

\(4{x}^{2}+16{y}^{2}=1\)

15

\(\frac{{(x-2)}^{2}}{49}+\frac{{(y-4)}^{2}}{25}=1\)

Show answer

\(\frac{{(x-2)}^{2}}{{7}^{2}}+\frac{{(y-4)}^{2}}{{5}^{2}}=1;\) Endpoints of major axis \((9,4),(-5,4).\) Endpoints of minor axis \((2,9),(2,-1).\) Foci at \((2+2\sqrt{6},4),(2-2\sqrt{6},4).\)

16

\(\frac{{(x-2)}^{2}}{81}+\frac{{(y+1)}^{2}}{16}=1\)

17

\(\frac{{(x+5)}^{2}}{4}+\frac{{(y-7)}^{2}}{9}=1\)

Show answer

\(\frac{{(x+5)}^{2}}{{2}^{2}}+\frac{{(y-7)}^{2}}{{3}^{2}}=1;\) Endpoints of major axis \((-5,10),(-5,4).\) Endpoints of minor axis \((-3,7),(-7,7).\) Foci at \((-5,7+\sqrt{5}),(-5,7-\sqrt{5}).\)

18

\(\frac{{(x-7)}^{2}}{49}+\frac{{(y-7)}^{2}}{49}=1\)

19

\(4{x}^{2}-8x+9{y}^{2}-72y+112=0\)

Show answer

\(\frac{{(x-1)}^{2}}{{3}^{2}}+\frac{{(y-4)}^{2}}{{2}^{2}}=1;\) Endpoints of major axis \((4,4),(-2,4).\) Endpoints of minor axis \((1,6),(1,2).\) Foci at \((1+\sqrt{5},4),(1-\sqrt{5},4).\)

20

\(9{x}^{2}-54x+9{y}^{2}-54y+81=0\)

21

\(4{x}^{2}-24x+36{y}^{2}-360y+864=0\)

Show answer

\(\frac{{(x-3)}^{2}}{{(3\sqrt{2})}^{2}}+\frac{{(y-5)}^{2}}{{(\sqrt{2})}^{2}}=1;\) Endpoints of major axis \((3+3\sqrt{2},5),(3-3\sqrt{2},5).\) Endpoints of minor axis \((3,5+\sqrt{2}),(3,5-\sqrt{2}).\) Foci at \((7,5),(-1,5).\)

22

\(4{x}^{2}+24x+16{y}^{2}-128y+228=0\)

23

\(4{x}^{2}+40x+25{y}^{2}-100y+100=0\)

Show answer

\(\frac{{(x+5)}^{2}}{{(5)}^{2}}+\frac{{(y-2)}^{2}}{{(2)}^{2}}=1;\) Endpoints of major axis \((0,2),(-10,2).\) Endpoints of minor axis \((-5,4),(-5,0).\) Foci at \((-5+\sqrt{21},2),(-5-\sqrt{21},2).\)

24

\({x}^{2}+2x+100{y}^{2}-1000y+2401=0\)

25

\(4{x}^{2}+24x+25{y}^{2}+200y+336=0\)

Show answer

\(\frac{{(x+3)}^{2}}{{(5)}^{2}}+\frac{{(y+4)}^{2}}{{(2)}^{2}}=1;\) Endpoints of major axis \((2,-4),(-8,-4).\) Endpoints of minor axis \((-3,-2),(-3,-6).\) Foci at \((-3+\sqrt{21},-4),(-3-\sqrt{21},-4).\)

26

\(9{x}^{2}+72x+16{y}^{2}+16y+4=0\)

For the following exercises, find the foci for the given ellipses.

27

\(\frac{{(x+3)}^{2}}{25}+\frac{{(y+1)}^{2}}{36}=1\)

Show answer

Foci \((-3,-1+\sqrt{11}),(-3,-1-\sqrt{11})\)

28

\(\frac{{(x+1)}^{2}}{100}+\frac{{(y-2)}^{2}}{4}=1\)

29

\({x}^{2}+{y}^{2}=1\)

Show answer

Focus \((0,0)\)

30

\({x}^{2}+4{y}^{2}+4x+8y=1\)

31

\(10{x}^{2}+{y}^{2}+200x=0\)

Show answer

Foci \((-10,30),(-10,-30)\)

Graphical

For the following exercises, graph the given ellipses, noting center, vertices, and foci.

32

\(\frac{{x}^{2}}{25}+\frac{{y}^{2}}{36}=1\)

33

\(\frac{{x}^{2}}{16}+\frac{{y}^{2}}{9}=1\)

Show answer

Center \((0,0),\) Vertices \((4,0),(-4,0),(0,3),(0,-3),\) Foci \((\sqrt{7},0),(-\sqrt{7},0)\)

34

\(4{x}^{2}+9{y}^{2}=1\)

35

\(81{x}^{2}+49{y}^{2}=1\)

Show answer

Center \((0,0),\) Vertices \((\frac{1}{9},0),(-\frac{1}{9},0),(0,\frac{1}{7}),(0,-\frac{1}{7}),\) Foci \((0,\frac{4\sqrt{2}}{63}),(0,-\frac{4\sqrt{2}}{63})\)

36

\(\frac{{(x-2)}^{2}}{64}+\frac{{(y-4)}^{2}}{16}=1\)

37

\(\frac{{(x+3)}^{2}}{9}+\frac{{(y-3)}^{2}}{9}=1\)

Show answer

Center \((-3,3),\) Vertices \((0,3),(-6,3),(-3,0),(-3,6),\) Focus \((-3,3)\)

Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.

38

\(\frac{{x}^{2}}{2}+\frac{{(y+1)}^{2}}{5}=1\)

39

\(4{x}^{2}-8x+16{y}^{2}-32y-44=0\)

Show answer

Center \((1,1),\) Vertices \((5,1),(-3,1),(1,3),(1,-1),\) Foci \((1+2\sqrt{3},1),(1-2\sqrt{3},1)\)

40

\({x}^{2}-8x+25{y}^{2}-100y+91=0\)

41

\({x}^{2}+8x+4{y}^{2}-40y+112=0\)

Show answer

Center \((-4,5),\) Vertices \((-2,5),(-6,5),(-4,6),(-4,4),\) Foci \((-4+\sqrt{3},5),(-4-\sqrt{3},5)\)

42

\(64{x}^{2}+128x+9{y}^{2}-72y-368=0\)

43

\(16{x}^{2}+64x+4{y}^{2}-8y+4=0\)

Show answer

Center \((-2,1),\) Vertices \((0,1),(-4,1),(-2,5),(-2,-3),\) Foci \((-2,1+2\sqrt{3}),(-2,1-2\sqrt{3})\)

44

\(100{x}^{2}+1000x+{y}^{2}-10y+2425=0\)

45

\(4{x}^{2}+16x+4{y}^{2}+16y+16=0\)

Show answer

Center \((-2,-2),\) Vertices \((0,-2),(-4,-2),(-2,0),(-2,-4),\) Focus \((-2,-2)\)

For the following exercises, use the given information about the graph of each ellipse to determine its equation.

46

Center at the origin, symmetric with respect to the x- and y-axes, focus at \((4,0),\) and point on graph \((0,3).\)

47

Center at the origin, symmetric with respect to the x- and y-axes, focus at \((0,-2),\) and point on graph \((5,0).\)

Show answer

\(\frac{{x}^{2}}{25}+\frac{{y}^{2}}{29}=1\)

48

Center at the origin, symmetric with respect to the x- and y-axes, focus at \((3,0),\) and major axis is twice as long as minor axis.

49

Center \((4,2)\) ; vertex \((9,2)\) ; one focus: \((4+2\sqrt{6},2)\) .

Show answer

\(\frac{{(x-4)}^{2}}{25}+\frac{{(y-2)}^{2}}{1}=1\)

50

Center \((3,5)\) ; vertex \((3,11)\) ; one focus: \((3,\,\text{5+4}\sqrt{\text{2}})\)

51

Center \((-3,4)\) ; vertex \((1,4)\) ; one focus: \((-3+2\sqrt{3},4)\)

Show answer

\(\frac{{(x+3)}^{2}}{16}+\frac{{(y-4)}^{2}}{4}=1\)

For the following exercises, given the graph of the ellipse, determine its equation.

52

53

Show answer

\(\frac{{x}^{2}}{81}+\frac{{y}^{2}}{9}=1\)

54

55

Show answer

\(\frac{{(x+2)}^{2}}{4}+\frac{{(y-2)}^{2}}{9}=1\)

56

Extensions

For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula \(\text{Area}=a\cdot b\cdot \pi .\)

57

\(\frac{{(x-3)}^{2}}{9}+\frac{{(y-3)}^{2}}{16}=1\)

Show answer

\(\text{Area = 12}\pi \,\text{square}\,\text{units}\)

58

\(\frac{{(x+6)}^{2}}{16}+\frac{{(y-6)}^{2}}{36}=1\)

59

\(\frac{{(x+1)}^{2}}{4}+\frac{{(y-2)}^{2}}{5}=1\)

Show answer

\(\text{Area = 2}\sqrt{\text{5}}\pi\) square units.

60

\(4{x}^{2}-8x+9{y}^{2}-72y+112=0\)

61

\(9{x}^{2}-54x+9{y}^{2}-54y+81=0\)

Show answer

\(\text{Area = 9}\pi\) square units.

Real-World Applications

62

Find the equation of the ellipse that will just fit inside a box that is 8 units wide and 4 units high.

63

Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high. Express in terms of \(h,\) the height.

Show answer

\(\frac{{x}^{2}}{4{h}^{2}}+\frac{{y}^{2}}{\frac{1}{4}{h}^{2}}=1\)

64

An arch has the shape of a semi-ellipse (the top half of an ellipse). The arch has a height of 8 feet and a span of 20 feet. Find an equation for the ellipse, and use that to find the height to the nearest 0.01 foot of the arch at a distance of 4 feet from the center.

65

An arch has the shape of a semi-ellipse. The arch has a height of 12 feet and a span of 40 feet. Find an equation for the ellipse, and use that to find the distance from the center to a point at which the height is 6 feet. Round to the nearest hundredth.

Show answer

\(\frac{{x}^{2}}{400}+\frac{{y}^{2}}{144}=1\) . Distance = 17.32 feet

66

A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center.

67

A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet, and the foci are located 30 feet from the center, find the height of the ceiling at the center.

Show answer

Approximately 51.96 feet

68

A person is standing 8 feet from the nearest wall in a whispering gallery. If that person is at one focus, and the other focus is 80 feet away, what is the length and height at the center of the gallery?

Glossary

center of an ellipse

the midpoint of both the major and minor axes

conic section

any shape resulting from the intersection of a right circular cone with a plane

ellipse

the set of all points \((x,y)\) in a plane such that the sum of their distances from two fixed points is a constant

foci

plural of focus

focus (of an ellipse)

one of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point \((x,y)\) on the ellipse is a constant

major axis

the longer of the two axes of an ellipse

minor axis

the shorter of the two axes of an ellipse