11.1 Distance and Midpoint Formulas; Circles

Learning objectives

Learning objectives

• Use the Distance Formula

• Use the Midpoint Formula

• Write the equation of a circle in standard form

• Graph a circle

In this chapter we will be looking at the conic sections, usually called the conics, and their properties. The conics are curves that result from a plane intersecting a double cone—two cones placed point-to-point. Each half of a double cone is called a nappe.

There are four conics—the circle, parabola, ellipse, and hyperbola. The next figure shows how the plane intersecting the double cone results in each curve.

Each of the curves has many applications that affect your daily life, from your cell phone to acoustics and navigation systems. In this section we will look at the properties of a circle.

Use the Distance Formula

We have used the Pythagorean Theorem to find the lengths of the sides of a right triangle. Here we will use this theorem again to find distances on the rectangular coordinate system. By finding distance on the rectangular coordinate system, we can make a connection between the geometry of a conic and algebra—which opens up a world of opportunities for application.

Our first step is to develop a formula to find distances between points on the rectangular coordinate system. We will plot the points and create a right triangle much as we did when we found slope in Graphs and Functions. We then take it one step further and use the Pythagorean Theorem to find the length of the hypotenuse of the triangle—which is the distance between the points.

Example 11.1

Solution

The method we used in the last example leads us to the formula to find the distance between the two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2}).$

When we found the length of the horizontal leg we subtracted $6-2$ which is ${x}_{2}-{x}_{1}.$

When we found the length of the vertical leg we subtracted $4-1$ which is ${y}_{2}-{y}_{1}.$

If the triangle had been in a different position, we may have subtracted ${x}_{1}-{x}_{2}$ or ${y}_{1}-{y}_{2}.$ The expressions ${x}_{2}-{x}_{1}$ and ${x}_{1}-{x}_{2}$ vary only in the sign of the resulting number. To get the positive value-since distance is positive- we can use absolute value. So to generalize we will say $|{x}_{2}-{x}_{1}|$ and $|{y}_{2}-{y}_{1}|.$

In the Pythagorean Theorem, we substitute the general expressions $|{x}_{2}-{x}_{1}|$ and $|{y}_{2}-{y}_{1}|$ rather than the numbers.

$\,{a}^{2}+{b}^{2}={c}^{2}$

Substitute in the values. · $\,{(\ · {x}_{2}-{x}_{1}\ · )}^{2}+{(\ · {y}_{2}-{y}_{1}\ · )}^{2}={d}^{2}$ Squaring the expressions makes them positive, so we eliminate the absolute value bars. · $\,{({x}_{2}-{x}_{1})}^{2}+{({y}_{2}-{y}_{1})}^{2}={d}^{2}$ Use the Square Root Property. · $\,d=\pm\sqrt{{({x}_{2}-{x}_{1})}^{2}+{({y}_{2}-{y}_{1})}^{2}}$ Distance is positive, so eliminate the negative value. · $\,d=\sqrt{{({x}_{2}-{x}_{1})}^{2}+{({y}_{2}-{y}_{1})}^{2}}$

This is the Distance Formula we use to find the distance d between the two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2}).$

Example 11.2

Solution

Write the Distance Formula. · $d=\sqrt{{({x}_{2}-{x}_{1})}^{2}+{({y}_{2}-{y}_{1})}^{2}}$ Label the points, $(\overset{{x}_{1},{y}_{1}}{-5,-3}),(\overset{{x}_{2},{y}_{2}}{7,2})$ and substitute. · $d=\sqrt{{(7-(-5))}^{2}+{(2-(-3))}^{2}}$ Simplify. · $d=\sqrt{{12}^{2}+{5}^{2}}$

$d=\sqrt{144+25}$

$d=\sqrt{169}$

$d=13$

Example 11.3

Solution

Write the Distance Formula. · $d=\sqrt{{({x}_{2}-{x}_{1})}^{2}+{({y}_{2}-{y}_{1})}^{2}}$ Label the points, $(\overset{{x}_{1},{y}_{1}}{10,-4}),(\overset{{x}_{2},{y}_{2}}{-1,5})$ and substitute. · $d=\sqrt{{(-1-10)}^{2}+{(5-(-4))}^{2}}$ Simplify. · $d=\sqrt{{(-11)}^{2}+{9}^{2}}$

$d=\sqrt{121+81}$

$d=\sqrt{202}$

Since 202 is not a perfect square, we can leave the answer in exact form or find a decimal approximation. · $\begin{array}{l} d=\sqrt{202} \\ \,\mathrm{or} \\ d\approx{}14.2 \\ \end{array}$

Use the Midpoint Formula

It is often useful to be able to find the midpoint of a segment. For example, if you have the endpoints of the diameter of a circle, you may want to find the center of the circle which is the midpoint of the diameter. To find the midpoint of a line segment, we find the average of the x-coordinates and the average of the y-coordinates of the endpoints.

Example 11.4

Solution

Both the Distance Formula and the Midpoint Formula depend on two points, $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2}).$ It is easy to confuse which formula requires addition and which subtraction of the coordinates. If we remember where the formulas come from, it may be easier to remember the formulas.

Write the Equation of a Circle in Standard Form

As we mentioned, our goal is to connect the geometry of a conic with algebra. By using the coordinate plane, we are able to do this easily.

We define a circle as 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.

This is the standard form of the equation of a circle with center, $(h,k),$ and radius, r.

Example 11.5

Solution

In the last example, the center was $(0,0).$ Notice what happened to the equation. Whenever the center is $(0,0),$ the standard form becomes ${x}^{2}+{y}^{2}={r}^{2}.$

Example 11.6

Solution

In the next example, the radius is not given. To calculate the radius, we use the Distance Formula with the two given points.

Example 11.7

Solution

The radius is the distance from the center to any point on the circle so we can use the distance formula to calculate it. We will use the center $(2,4)$ and point $(-2,1)$

Use the Distance Formula to find the radius. · $\,r=\sqrt{{({x}_{2}-{x}_{1})}^{2}+{({y}_{2}-{y}_{1})}^{2}}$ Substitute the values. $(\overset{{x}_{1},{y}_{1}}{2,4}),(\overset{{x}_{2},{y}_{2}}{-2,1})$ · $\,r=\sqrt{{(-2-2)}^{2}+{(1-4)}^{2}}$ Simplify. · $\,r=\sqrt{{(-4)}^{2}+{(-3)}^{2}}$

$\,r=\sqrt{16+9}$

$\,r=\sqrt{25}$

$\,r=5$

Now that we know the radius, $r=5,$ and the center, $(2,4),$ we can use the standard form of the equation of a circle to find the equation.

Use the standard form of the equation of a circle. · $\,{(x-h)}^{2}+{(y-k)}^{2}={r}^{2}$ Substitute in the values. · $\,{(x-2)}^{2}+{(y-4)}^{2}={5}^{2}$ Simplify. · $\,{(x-2)}^{2}+{(y-4)}^{2}=25$

Graph a Circle

Any equation of the form ${(x-h)}^{2}+{(y-k)}^{2}={r}^{2}$ is the standard form of the equation of a circle with center, $(h,k),$ and radius, r. We can then graph the circle on a rectangular coordinate system.

Note that the standard form calls for subtraction from x and y. In the next example, the equation has $x+2,$ so we need to rewrite the addition as subtraction of a negative.

Example 11.8

Solution

To find the center and radius, we must write the equation in standard form. In the next example, we must first get the coefficient of ${x}^{2},{y}^{2}$ to be one.

Example 11.9

Solution

If we expand the equation from Example 11.8, ${(x+2)}^{2}+{(y-1)}^{2}=9,$ the equation of the circle looks very different.

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

Square the binomials. · ${x}^{2}+4x+4+{y}^{2}-2y+1=9$ Arrange the terms in descending degree order, and get zero on the right · ${x}^{2}+{y}^{2}+4x-2y-4=0$

This form of the equation is called the general form of the equation of the circle.

If we are given an equation in general form, we can change it to standard form by completing the squares in both x and y. Then we can graph the circle using its center and radius.

Example 11.10

Solution

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

In the next example, there is a y-term and a ${y}^{2}$-term. But notice that there is no x-term, only an ${x}^{2}$-term. We have seen this before and know that it means h is 0. We will need to complete the square for the y terms, but not for the x terms.

Example 11.11

Solution

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

Section 11.1 Exercises

Section 11.1 Exercises

Use the Distance Formula

In the following exercises, find the distance between the points. Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.

Use the Midpoint Formula

In the following exercises, ⓐ find the midpoint of the line segment whose endpoints are given and ⓑ plot the endpoints and the midpoint on a rectangular coordinate system.

Write the Equation of a Circle in Standard Form

In the following exercises, write the standard form of the equation of the circle with the given radius and center $(0,0).$

In the following exercises, write the standard form of the equation of the circle with the given radius and center

For the following exercises, write the standard form of the equation of the circle with the given center with point on the circle.

Graph a Circle

In the following exercises, ⓐ find the center and radius, then ⓑ graph each circle.

In the following exercises, ⓐ identify the center and radius and ⓑ graph.

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

1. Use the rectangular coordinate system to find the distance between the points $(6,1)$ and $(2,-2).$

2. Use the rectangular coordinate system to find the distance between the points $(5,3)$ and $(-3,-3).$

3. Use the Distance Formula to find the distance between the points $(-4,-5)$ and $(5,7).$

1. Use the Midpoint Formula to find the midpoint of the line segment whose endpoints are $(-3,-5)$ and $(5,7).$ Plot the endpoints and the midpoint on a rectangular coordinate system.

2. Use the Midpoint Formula to find the midpoint of the line segment whose endpoints are $(-2,-5)$ and $(6,-1).$ Plot the endpoints and the midpoint on a rectangular coordinate system.

3. Write the standard form of the equation of the circle with a radius of 6 and center $(0,0).$

1. Write the standard form of the equation of the circle with a radius of 8 and center $(0,0).$

2. Write the standard form of the equation of the circle with a radius of 7 and center $(2,-4).$

3. Write the standard form of the equation of the circle with a radius of 9 and center $(-3,-5).$

1. $(3,-1)$ and $(1,7)$

2. $(0,-5)$ and $(4,-3)$

3. $(3,-1)$ and $(4,-2)$

1. Radius: 7

2. Radius: $\sqrt{2}$

3. Radius: 1, center: $(3,5)$

1. Radius: $2.5,$ center: $(1.5,-3.5)$

2. Center $(3,-2)$ with point $(3,6)$

3. Center $(4,4)$ with point $(2,2)$

1. ${(x+5)}^{2}+{(y+3)}^{2}=1$

2. ${(x-4)}^{2}+{(y+2)}^{2}=16$

3. ${x}^{2}+{(y+2)}^{2}=25$

1. ${(x-1.5)}^{2}+{(y+2.5)}^{2}=0.25$

2. ${x}^{2}+{y}^{2}=64$

3. $2{x}^{2}+2{y}^{2}=8$

1. ${x}^{2}+{y}^{2}+2x+6y+9=0$

2. ${x}^{2}+{y}^{2}-4x+10y-7=0$

3. ${x}^{2}+{y}^{2}+6y+5=0$

1. Use the Distance Formula to find the distance between the points $(-2,-5)$ and $(-14,-10).$

2. Use the Distance Formula to find the distance between the points $(-4,-5)$ and $(3,4).$ Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.

3. Use the Distance Formula to find the distance between the points $(-2,-5)$ and $(-3,-4).$ Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.

1. Write the standard form of the equation of the circle with center $(2,1)$ that also contains the point $(-2,-2).$

2. Write the standard form of the equation of the circle with center $(7,1)$ that also contains the point $(-1,-5).$

3. ⓐ Find the center and radius, then ⓑ graph the circle: ${(x-3)}^{2}+{(y+4)}^{2}=4.$

1. ⓐ Find the center and radius, then ⓑ graph the circle: ${(x-3)}^{2}+{(y-1)}^{2}=16.$

2. ⓐ Find the center and radius, then ⓑ graph the circle: $3{x}^{2}+3{y}^{2}=27$

3. ⓐ Find the center and radius, then ⓑ graph the circle: $5{x}^{2}+5{y}^{2}=125$

1. ⓐ Find the center and radius, then ⓑ graph the circle: ${x}^{2}+{y}^{2}-6x-8y+9=0.$

2. ⓐ Find the center and radius, then ⓑ graph the circle: ${x}^{2}+{y}^{2}+6x-2y+1=0.$

3. ⓐ Find the center and radius, then ⓑ graph the circle: ${x}^{2}+{y}^{2}-2x-3=0.$

1. ⓐ Find the center and radius, then ⓑ graph the circle: ${x}^{2}+{y}^{2}-12y+11=0.$

2. $(2,0)$ and $(5,4)$

3. $(-3,-5)$ and $(0,1)$

1. ${x}^{2}+{y}^{2}+4x=0$

2. Explain the relationship between the distance formula and the equation of a circle.

3. In your own words, state the definition of a circle.