Ellipse

Equation of the ellipse, standard equation of the ellipse

Major axis, minor axis, and vertices

The focal parameter, latus rectum

Equations of the ellipse examples

Equation of the ellipse, standard equation of the ellipse

f in the direction of axes we introduce a coordinate system so that the center of the ellipse coincides with
he
origin, then coordinates of foci are

F1(c, 0) and F2( c, 0).

For every point P(x, y) of the ellipse, according to

definition r1 + r2 = 2a, it follows that

after squaring

and reducing

Repeated squaring and grouping gives

(a2  c2) · x2 + a2y2 = a2 · (a2  c2),

and since a2  c2 = b2
 follows b2x2 + a2y2 = a2b2 equation of the ellipse,

and after division by a2b2, standard equation of the ellipse.

t follows from the equation that an ellipse is defined by values of a and b, or as they are associated
hrough
he relation a2  c2 = b2, we can say that it is defined by any pair of these three quantities.

Intersections of an ellipse and the coordinate axes we determine from

by putting,
equation

y = 0 => x = + a, so obtained are vertices at the ends of the major axis A1(a, 0) and A2(a, 0), and

x = 0 => y = + b, obtained are co-vertices, the endpoints of the minor axis B1(0, b) and B2(0, b).
The line segments A1A2 = 2a and B1B2 = 2b are the major and minor axes while a and b are the
semi-major and semi-minor axes respectively. So the arc of the radius a centered at B1and B2 intersects
he major axis at the foci F1and F2.

The focal parameter, called latus rectum and denoted 2p, is the chord perpendicular to the major axis
passing through any of the foci, as shows the above figure. The length of which equals the absolute value
of ordinates of the points of the ellipse whose abscissas x = c or x = c that is

so the length of the latus

the length of the semi-latus rectum of the ellipse.
rectum

Equations of the ellipse examples

Example: Given is equation of the ellipse 9x2 + 25y2 = 225, find the lengths of semi-major and semi-
minor axes, coordinates of the foci, the eccentricity and the length of the semi-latus rectum.

Solution: From the standard equation we can find the semi-axes lengths dividing the given
equation by 225,

coordinates of the foci F1(c, 0) and F2( c, 0), since

Example: From given quantities of an ellipse determine remaining unknown quantities and write equation
of the ellipse,

Solution: a) Using

therefore, the semi-minor axis

the linear eccentricity

the semi latus rectum and the equation of the ellipse

the eccentricity and the equation of the ellipse

 the semi latus rectum and the equation of the ellipse

d) unknown quantities expressed through given values,

Example: Find the equation of the ellipse whose focus is F2(6, 0) and which passes through the point
A(53, 4).

Solution: Coordinates of the point A(53, 4) must satisfy equation of the ellipse, therefore

thus, the equation of the ellipse

Example: Write equation of the ellipse passing through points A(4, 2) and B(8, 1).
Solution: Given points must satisfy equation of the ellipse, so

Therefore, the equation of the ellipse or x2 + 16y2 = 80.

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Boundless Algebra
Conic Sections

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The Circle and the Ellipse

Introduction to Circles
The equation for a circle is an extension of the distance formula.

Learning Objectives

Explain how the equation of a circle describes its properties

Key Takeaways

Key Points

 A circle is defined as the set of points that are a fixed distance from a center point.
 The distance formula can be extended directly to the definition of a circle by noting
that the radius is the distance between the center of a circle and the edge.
 The general equation for a circle, centered at (a,b)

with radius r, is the set of all points (x,y) such that (x−a)2+(y−b)2=r2
 .
 Pi (π

 ) is the ratio of a circle’s circumference to its diameter.

Key Terms

 diameter: Two times the radius of a circle.

 area: The interior surface of a circle, given by A=πr2

 .
 circle: A two-dimensional geometric figure, consisting of the set of all those points in
a plane that are equally distant from another point.
 radius: A line segment between any point on the circumference of a circle and its
center.

The definition of a circle is as simple as the shape. A circle is the set of all points that are at a
certain distance from a center point. This definition is what gives us the concept of the radius
of a circle, which is equal to that certain distance.

Since we know a circle is the set of points a fixed distance from a center point, let’s look at
how we can construct a circle in a Cartesian coordinate plane with variables x

and y. To find a formula for this, suppose that the center is the point (a,b). According to the
distance formula, the distance c from the point (a,b) to any other point (x,y)

is:

c=√ (x−a)2+(y−b)2
If we now square this equation on both sides, we have:

c2=(x−a)2+(y−b)2

Remember that the distance between the center (a,b)

and any point (x,y) on the circle is that fixed distance, which is called the radius. So let’s
change this equation so that it uses r instead of c

r2=(x−a)2+(y−b)2

This is the general formula for a circle with center (a,b)

and radius r

. Notice that all we have done is slightly rearrange the distance formula equation.

Graph of a circle: The circle with center (a,b)

is graphed in the Cartesian plane.

Parts of a Circle

Now that we have an algebraic foundation for the circle, let’s connect it to what we already
know about some different parts of the circle.

Diameter

The diameter is any straight line that passes through the center of the circle. It is equal to
twice the radius, so:

d=2r
Circumference

The circumference is the length of the path around the circle. Algebraically it is given by:

c=2πr

or equivalently by c=πd

. The number π

(pi) is defined by this relationship. It is the ratio of any circle’s circumference to its own
diameter.

Area

The area of a circle is given by:

A=πr2

Notice that the radius is the only defining parameter for the size of any particular circle, and
so it is the only variable that the area depends on.

Introduction to Ellipses
Understand the equation of an ellipse as a stretched circle.

Learning Objectives

Connect the equation for an ellipse to the equation for a circle with stretching factors

Key Takeaways

Key Points

 An ellipse and a circle are both examples of conic sections.

 A circle is a special case of an ellipse, with the same radius for all points.
 By stretching a circle in the x or y direction, an ellipse is created.

An ellipse is one of the shapes called conic sections, which is formed by the intersection of a
plane with a right circular cone. The general equation of an ellipse centered at (h,k)

is:

(x−h)2a2+(y−k)2b2=1

when the major axis of the ellipse is horizontal.

Ellipse: An ellipse is a conic section, formed by the intersection of a plane with a right
circular cone.

In most definitions of the conic sections, the circle is defined as a special case of the ellipse,
when the plane is parallel to the base of the cone. However, it is also possible to begin with
the definition of a circle and use graphical transformations to arrive at the general formula for
an ellipse.

Recall that a circle is defined as the set of all points equidistant from a common center. For
simplicity, we will choose that center to be (0,0)

, the origin of the x–y

plane. Then we can write the equation of the circle in this way:

x2+y2=r2

In this equation, r

is the radius of the circle. A circle has only one radius—the distance from the center to any
point is the same. To change our circle into an ellipse, we will have to stretch or squeeze the
circle so that the distances are no longer the same. First, let’s start with a specific circle that’s
easy to work with, the circle centered at the origin with radius 1

x2+y2=1

To make this into an ellipse, we must distort the circular shape so that it is no longer
symmetric between x
and y. To do this, we introduce a scaling factor into one or both of the
x–y coordinates. Let’s start by dividing all x coordinates by a factor a, and therefore scaling
the x values. We simply substitute xa into the equation instead of x. Important note: We
assume that a>1

(xa)2+y2=1

Ellipse along x

-axis: The ellipse (x3)2+y2=1 has been stretched along the x-axis by a factor of 3 as
compared to the circle x2+y2=1

Every x

-value that solved the old equation must now be multiplied by a in order to solve the new
equation. This has the effect of stretching the ellipse further out on the x-axis, because larger
values of x
are now the solutions.

Similarly, we can scale all the values of y

by a factor b (we also assume b>1

).

x2+(yb)2=1

Now all the y

values are stretched vertically, further away from the origin.

Ellipse along y

-axis: The ellipse x2+(y3)2=1 has been stretched along the y-axis by a factor of 3 as
compared to the circle x2+y2=1

.
If we stretch in both the x

and y

directions and distribute the powers of two through the parentheses, we get:

x2a2+y2b2=1

which is exactly the equation of a horizontal ellipse centered at the origin.

If we had used scaling factors that were less than one, it would have compressed the shape
instead of stretching it further out.

Parts of an Ellipse
Ellipses are one of the types of conic sections. The specific features of an ellipse can be
determined from its equation.

Learning Objectives

Discuss how the equation of an ellipse describes its properties

Key Takeaways

Key Points

 An ellipse is formed by a plane intersecting a cone at an angle to its base.

 All ellipses have two focal points, or foci. The sum of the distances from every point
on the ellipse to the two foci is a constant.
 All ellipses have a center and a major and minor axis.
 All ellipses have eccentricity values greater than or equal to zero, and less than one.

Equation of an Ellipse

An ellipse is a conic section, formed by the intersection of a plane with a right circular cone.
The standard form for the equation of the ellipse is:

(x−h)2a2+(y−k)2b2=1

if the ellipse is oriented horizontally, and:

(y−k)2a2+(x−h)2b2=1

if the ellipse is oriented vertically. We will use the horizontal case to demonstrate how to
determine the properties of an ellipse from its equation, so that a

is associated with x-coordinates, and b

 with y-coordinates. For a vertical ellipse, the association is reversed.

Diagram of an ellipse: This diagram of a horizontal ellipse shows the ellipse itself in red, the
center C

at the origin, the focal points at (+f,0) and (−f,0), the major axis vertices at (+a,0) and
(−a,0), the minor axis vertices at (0,+b) and (0,−b). It also shows how the sum of the
distances from any point on the ellipse to the two foci is a constant at 2a (the length of the
major axis), and how the eccentricity is determined by the ratio of the distance from a point
on the ellipse to one of the foci to the perpendicular distance from the point to a line D

called the directrix, outside the ellipse and perpendicular to the major axis.

Parts of an Ellipse

Center

The center of the ellipse has coordinates (h,k)

Major Axis

The major axis of the ellipse is the longest width across it. For a horizontal ellipse, that axis is
parallel to the x

-axis. The major axis has length 2a. Its endpoints are the major axis vertices, with coordinates
(h±a,k)
.

Minor Axis

The minor axis of the ellipse is the shortest width across it. For a horizontal ellipse, it is
parallel to the y

-axis. The minor axis has length 2b. Its endpoints are the minor axis vertices, with
coordinates (h,k±b)

Foci

The foci are two points inside the ellipse that characterize its shape and curvature. For a
horizontal ellipse, the foci have coordinates (h±c,k)

, where the focal length c

is given by

c2=a2−b2

Eccentricity

All conic sections have an eccentricity value, denoted e

. All ellipses have eccentricities in the range 0≤e<1. An eccentricity of zero is the special
case where the ellipse becomes a circle. An eccentricity of 1

is a parabola, not an ellipse.

The eccentricity is defined as:

e=ca

or, equivalently:

e=√ a2−b2 a=√ a2−b2a2 =√ 1−b2a2

The orbits of the planets and their moons are ellipses with very low eccentricities, which is to
say they are nearly circular. The orbits of comets around the sun can be much more eccentric.
For comets and planets, the sun is located at one focus of their elliptical orbits.

Applications of Circles and Ellipses

Circles and ellipses are encountered in everyday life, and knowing how to solve their
equations is useful in many situations.

Learning Objectives

Describe applications of the circle and ellipse formulas to real world problems

Key Takeaways

Key Points

 The standard form of the equation for a circle is (x−h)2+(y−k)2=r2

, where r is the radius, and (h,k)

 is the center point coordinate.
 The standard form of the equation for an ellipse is (x−h)2a2+(y−k)2b2=1, where (h,k) is
the center point coordinate, 2a is the length of the major/ minor axis, and 2b
 is the minor/major axis length.
 If a>b, the major axis is parallel to the x axis. If a<b, the major axis is parallel to the y axis
(perpendicular to the x

 axis).
 To determine the parameters of a circle or an ellipse, you must first put the equation
into the standard form.

Key Terms

 ellipse: A closed curve, the locus of a point such that the sum of the distances from
that point to two other fixed points (called the foci of the ellipse) is constant;
equivalently, the conic section that is the intersection of a cone with a plane that does
not intersect the base of the cone.
 circle: A two-dimensional geometric figure, consisting of the set of all those points in
a plane that are equally distant from another point.

Circles are all around you in everyday life, from tires on cars to buttons on coats, as well as
on the tops of bowls, glasses, and water bottles. Ellipses are less common. One example is the
orbits of planets, but you should be able to find the area of a circle or an ellipse, or the
circumference of a circle, based on information given to you in a problem. Circles and
ellipses are examples of conic sections, which are curves formed by the intersection of a
plane with a cone.

Some sample problems are shown below, with solutions worked out.

Exercise 1

Let’s say you are a gardener, and you have just planted a lot of flowers that you want to
water. The flower bed is 15 feet wide, and 15 feet long. You are using a circular sprinkler
system, and the water reaches 6 feet out from the center. The sprinkler is located, from the
bottom left corner of the bed, 7 feet up, and 6 feet over.
 1. If the flower bed was a graph with the bottom left corner being the origin, what would
the equation of the circle be?
2. What is the area being watered by the sprinkler?
3. What percentage of the garden that is being watered?

Answer

If the bottom left corner is the origin, it has coordinates (0,0)

. From there the sprinkler is 6 feet over, so the x-coordinate of the center is 6. The sprinkler is
7 feet up, so the y-coordinate of the center is 7. Ergo, the center of the circle is at coordinate
(6,7)

. The water reaches 6 feet out from the sprinkler, so the circle radius is 6 feet. Therefore the
equation of this circle is:

(x−6)2+(y−7)2=36

The area that is watered by the sprinkler can be labeled Asprinkler

, and is:

Asprinkler=π⋅r2=π⋅62=36π

The first step to finding the percentage of the garden that is being watered is to check that
none of the water is falling outside the garden. The sprinkler is at coordinate (6,7)

, and the radius of the sprinkler is 6 feet. You can test that the water does not fall outside the
15×15

garden.

Once we know that the area that is watered is completely inside the garden, the percentage of
the garden that is watered can be found by dividing the area watered by the total area of the
garden, and then multiplying by 100%

Percentagewatered=AsprinklerAflowerbed⋅100%=36π152⋅100%=113.1225⋅100%=50.3%

Exercise 2

Now, let’s take it the other way. (x−4)2+(y+8)2=49

is the equation for a circle.

1. What are the coordinates of the center of the circle?

2. What is the radius?
3. Draw the circle.
 4. Find two points on the circle and plug them into the equation to make sure your
drawing is correct.

Answer

The center of the circle can be found by comparing the equation in this exercise to the
equation of a circle:

(x−h)2+(y−k)2=r2

The center of the circle is at coordinate (h,k)

, and so the center of the circle in this exercise is at (4,−8)

The radius of the circle is r

. From the equation of a circle, r2=49

. Therefore:

r2=49r=√ 49 =7
Exercise 2 circle: Graph of the circle (x–4)2+(y+8)2=49

There are many points you could choose. For example, the point (4,−1)

is at the top of the circle. Plugging this into the equation, we get:

(x−h)2+(y−k)2=r2(4−4)2+(−1+8)2=49(0)2+(7)2=4949=49

The left side is equal to the right side of the equation, and so this is a valid point on the circle.

The leftmost point on the circle is (−3,−8)

. Plugging this in to the equation:

(x−h)2+(y−k)2=r2(−3−4)2+(−8+8)2=49(−7)2+(0)2=4949=49

The left side is equal to the right side of the equation, and so this is a valid point on the circle.

Exercise 3
 1. Put 2x2+2y2+8x+24y+60=0

into the standard circle form: (x−h)2+(y−k)2=r2

1. .
2. What is the center and the radius of the circle?
3. Draw the circle.
4. Find two points on the circle and plug them into the equation to make sure your
drawing is correct.

Answer

First, divide the equation by the coefficient of x2

and y2, which is 2

2x2+2y2+8x+24y+60=02x2+2y2+8x+24y+602=022x22+2y22+8x2+24y2+602=02x2+y
2+4x+12y+30=0

Next, collect x

and y

terms together, and bring the number to the right side of the equation:

(x2+4x)+(y2+12y)=−30

Now, complete the square in both parentheses, subtracting or adding the necessary constant
to both sides of the equation:

(x2+4x+4)+(y2+12y+36)=−30+4+36(x2+4x+4)+(y2+12y+36)=10

Notice that each term is a perfect square, which gives:

(x+2)2+(y+6)2=10

This is now in the standard form for the equation of a circle.

The center of the circle is at coordinate (h,k)

, and so the center of the circle in this exercise is at (−2,−6)

The radius of the circle is r

. From the equation of a circle, r2=10

. Therefore:

r2=10r=√ 10

It is fine to leave the value of r

in this form.

Exercise 3 circle: Graph of the circle 2x2+2y2+8x+24y+60=0

There are many points you could choose. For example, (−5,−5)

and (−3,−9)

(−5+2)2+(−5+6)2=10(−3)2+(1)2=109+1=1010=10
(−3+2)2+(−9+6)2=10(−1)2+(−3)2=101+9=1010=10

Both are valid points on the circle.

Exercise 4

1. Put the equation 4x29+25y2=1

1. into standard form.

2. What is the center?
3. How long is the major axis?
4. How long is the minor axis?
5. Graph it.

Answer

This almost looks like an ellipse in standard form, doesn’t it? It even has a number on the
right side. But it isn’t. There is no room in the standard form for the values 4

and 25

in the numerators. How can we get rid of them to get into standard form?

Rewrite the left-hand term, 4x29

, by dividing the top and bottom of the fraction by 4

. Leave the bottom as a fraction; don’t make it a decimal.

Rewrite the right-hand term, 25y2

, as 25y21, and divide the top and bottom of the fraction by 25

. Leave the bottom as a fraction; don’t make it a decimal.

The equation is now:

4x29+25y2=14x2494+25y225125=1x294+y2125=1

From the standard equation, (x−h)2a2+(y−k)2b2=1

, we know that the center is at (h,k). Since these are both zero in this equation, the center is at
(0,0)

The major axis depends which is longer, a

or b

. Let’s solve for both, and find out which is larger afterward.

a2=94a=322a=62=3

b2=125b=152b=25

Since 3>25

, 2a>2b, and 2a is the major axis. The major axis length is 2a=3, and the minor axis length is
2b=25

Exercise 4 ellipse: Graph of the ellipse 4x29+25y2=1

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