Advanced Math
Chapter 8: Notes
Distance & Midpoint Formulas
Circles
Name:
April 15, 2013
DISTANCE FORMULA:
Example 1:
Find the distance between P1 (−3, 4) and P2 (5, 2) and simplify the radical if necessary
MIDPOINT:
Example 2:
Find the midpoint of the line segment between P1 (−2, 6) and P2 (3, 4) and reduce the fractions if
necessary
STANDARD FORM OF A CIRCLE: The standard form of a circle with center (ℎ, 𝑘) and radius 𝑟 is given by the equation:
Example 2:
(a)
Graph the circle with center (−1, 2) and radius 3
(b)
Write the equation of the circle in standard form
(c)
Expand the equation in part (b) to write the equation
in general form
Example 3:
Given the circle with a general form of 𝑥 2 + 𝑦 2 + 8𝑥 + 4𝑦 − 5 = 0
(a)
Write the equation of the circle in standard form
(b)
Identify the center and the radius
Center =
Example 4:
Radius =
Given the circle with a general form of 𝑥 2 + 𝑦 2 − 10𝑥 − 3 = 0
(a)
Write the equation of the circle in standard form
(b)
Identify the center and the radius
Center =
Radius =
Advanced Math
HOMEWORK
Distance & Midpoint Formulas
Circles
Name:
April 15, 2013
Find the distance between 𝐏𝟏 and 𝐏𝟐 and simplify the radical if necessary.
1. P1 (0, 4) and P2 (8, −11)
2. P1 (−5, 8) and P2 (−10, 14)
1.)_________________
2.)_________________
Find the midpoint of the line segment between 𝐏𝟏 and 𝐏𝟐 and reduce the fractions if necessary
3. P1 (0, 4) and P2 (8, −11)
4. P1 (−5, 8) and P2 (−10, 14)
3.)_________________
4.)_________________
Find the equation of the circle that satisfies the given conditions. Write your answer in standard form.
5. Center: (4, −3)
Radius: 7
6. Center: (0, 4)
Radius: √3
5.)______________________________
6.)______________________________
1
7. Center: (2 ,
1
)
4
Radius: √6
8. Center: (−2, 0)
Radius: 12
7.)______________________________
8.)______________________________
For the following equations in general form, (a) write the equation of the circle in standard form, (b) identify the
center, and (c) identify the radius.
9. 𝑥 2 + 𝑦 2 − 8𝑥 + 14𝑦 − 26 = 0
10. 𝑥 2 + 𝑦 2 + 12𝑥 − 10𝑦 + 18 = 0
11. 𝑥 2 + 𝑦 2 − 16𝑥 − 2𝑦 + 11 = 0
9a.)____________________________________
9b.)
Center:__________________________
9c.)
Radius:__________________________
10a.)___________________________________
10b.)
Center:__________________________
10c.)
Radius:__________________________
11a.)___________________________________
11b.)
Center:__________________________
11c.)
Radius:__________________________
12. 𝑥 2 + 𝑦 2 + 2𝑥 − 4𝑦 − 15 = 0
12a.)___________________________________
12b.)
Center:__________________________
12c.)
Radius:__________________________
A circle has a diameter with endpoints (2, 3) and (−4, 11).
What is the center of the circle?
13a.)_______________
Use today’s notes to help answer the following questions.
13.
14.
(a)
(b)
What formula did you use from today’s notes to find this?
(a)
A circle has a center at (2, 3) and goes through the point (−4, 11).
What is the length of the radius of the circle?
(b)
What formula did you use from today’s notes to find this?
13b.)____________________________
14a.)_______________
14b.)____________________________
Answers to HW:
1.
17
2.
√61
3.
(4, − 2)
4.
(−
5.
(𝑥 − 4)2 + (𝑦 + 3)2 = 49
6.
𝑥 2 + (𝑦 − 4)2 = 3
7.
(𝑥 − 2) + (𝑦 − 4) = 6
8.
(𝑥 + 2)2 + 𝑦 2 = 144
9a.
(𝑥 − 4)2 + (𝑦 + 7)2 = 91
9b.
(4, −7)
9c.
√91
10a.
(𝑥 + 6)2 + (𝑦 − 5)2 = 43
10b.
(−6, 5)
10c.
√43
11a.
(𝑥 − 8)2 + (𝑦 − 1)2 = 54
11b.
(8, 1)
11c.
3√6
12a.
(𝑥 + 1)2 + (𝑦 − 2)2 = 20
12b.
(−1, 2)
12c.
2√5
13a.
(−1, 7)
13b.
midpoint formula
14a.
10
14b.
distance formula
7
15
,
2
11)
1 2
1 2