圓方程式 Equations of Circles

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圓方程式 Equations of Circles

第 1 節 1st Period

Material Note

Vocabulary: Compass (圓規), Standard Equation of a

Circle (圓的標準式), Center (中心), Radius (半徑), Distance Formula (距離公式), Diameter (直徑).

Sentences:

1. A circle is the set of all points in a plane that are a fixed distance from a given point called the center of a circle. (平面上，和一個定點等距離的所有點所成的圖形稱為圓。這個定點稱為圓心。) 2. The distance from the center to a point on the

circle is called the radius of a circle. (圓心和圓上一點的距離稱為半徑。)

3. Let r be the radius of the circle C. (設 r 是圓 C 的半徑。)

4. Let point P(x, y) be any point on the circle C. (設 P(x, y)是圓 C 上的任意點。)

5. By the Distance Formula, you can get… (由距離公式可得… )

6. We call this



^{x h}^

 

²^ ^{y k}^



²^^r²^{is the}

Standard Equation of a Circle.

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Video: BYJU'S - Circles : Introduction

https://youtu.be/m9dpeG2rKdY

Vocabulary: Diameter (直徑), Chord (弦), Arc (弧),

Sector (扇形), Segment (弓形), Circumference (圓周), Exterior (外部), Interior (內部).

Translations:

The standard equation of a circle with center at M(h, k) and radius r is



^{x h}^

 

²^ ^{y k}^



²^^r²^.

Translations:

Example 1

Find the equation of the circle:

(1) The circle with the center



2, 3



and a radius 4.

(2) The circle with the center M



2, 3



and passes

through point A



5,1



. Solution

(1) From the standard equation of a circle, we know that



^x^²



²^



^y^{ }

 

³



²^⁴²

So that the equation of a circle is



^x^²



²^



^y^³



²^¹⁶

(2) The radius of the circle is



^{5 2}



²



¹

 

³



² ⁵

rAM     

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From the standard equation of a circle, we get the equation of a circle is



^x^²



²^



^y^³



²^²⁵

Vocabulary: General Form of a Circle (圓的一般式),

Quadratic Equation in Two Variables (二元二次方程式)

Translations:

Expand the standard form of a circle



^{x h}^



²^



^{y k}^



²^^r²

and we get

2 2 2 2 2

2 2 0

x y  hx ky h k r  .

This form of equation is similar to quadratic equation in two variables

2 2

+ 0

x y dx ey f 

, which is the “General Form of a Circle”.

General Form of a Circle

All of the equations of circles can be expressed as the form of two-variable quadratic equations:

2 2

0 x y dx ey  f . Translations:

Example 3

Find the center and the radius of the circle

2 2

: 2 6 6 0

C x y  x y  Solution

Complete the square for the x terms, and similarly for

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the y terms, and we get



^x²^²^x^¹

 

^ ^y²^⁶^y^⁹



^{   }^{6 1 9}

so that



^x^¹



²^



^y^³



² ^²²

From the standard equation of a circle we know that the center of the circle is



^{1, 3}^



and the radius is 2.

Vocabulary: Interior Point (內部點), Exterior Point (外 部點).

Illustrations:

Look at figure 5, point M is the center of the circle.

The radius of the circle is r and a point P.

(1) If point P lies on the circle, then MP r . (2) If point P lies exterior of the circle, then MP r . (3) If point P lies interior of the circle, then MP r . Example 8

Determine if these the three points ^P



^6,0



^,



^{2, 1}



Q   and ^R



^0,2



lie on inside, outside or on

the circle of this equation: ^C^:



^x^²



²^



^y^³



²^²⁵^.

Solution

From standard form of a circle we know the center is



^{2, 3}



M  , and the radius is 5.

Calculate the distance from point P, Q and R to center

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5



2, 3



M  respectively, and we get



^{6 2}



²



^{0 3}



² ⁵

PM     



^{2 2}



²



^{1 3}



² ^{20 5}

QM        



^{0 2}



²



^{2 3}



² ^{29 5}

RM      

Thus, point P is on the circle, point Q is inside of the circle and point R is outside of the circle.

補充題 Material 1

A city’s commuter system has three zones. Zone 1 serves people living within 3 miles of the city’s center. Zone 2 serves those between 3 and 7 miles (included) from the center. Zone 3 serves those over 7 miles from the center. (Shown in the right figure.)

Determine which zone serves people whose homes are

represented by the points ^{A 3,4}

 

^,^B



^6,5



^, ^C



^1,2



^, ^D



^0,3



^and ^E



^1,6



^.

Solution

We set the center of the city as ^O



^0,0



. Calculate the distance from point A, B, C, D and E to the center O respectively.



3 0



²



4 0



² 5

AO      , and 3AO7. A is in zone 2.



6 0



²



5 0



² 61

BO      , and BO  . B is in zone 3. 7



1 0



²



2 0



² 5

CO      , and CO  . C is in zone 1. 3



0 0



²



3 0



² 3

DO      , and DO  . D is in zone 2. 3



1 0



²



6 0



² 37

EO      , and 3EO7. E is in zone 2.

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Material 2

The epicenter of an earthquake is the point on Earth’s surface directly above the earthquake’s origin. A seismograph can be used to determine the distance to the epicenter of an earthquake.

Seismographs are needed in three different places to locate an earthquake’s epicenter.

Use the seismograph readings from locations A, B, and C to find the epicenter of an earthquake.

• The epicenter is 7 miles away from A(−2, 2.5).

• The epicenter is 4 miles away from B(4, 6).

• The epicenter is 5 miles away from C(3, −2.5).

Solution

The set of all points equidistant from a given point is a circle, so the epicenter is located on each of the following circles.

⊙A with center (−2, 2.5) and radius 7

⊙B with center (4, 6) and radius 4

⊙C with center (3, −2.5) and radius 5

To find the epicenter, graph the circles on a coordinate plane where each unit corresponds to one mile. Find the point of intersection of the three circles.

The epicenter is at about (5, 2).

Note

Word: Commuter (通勤者), Serve (服務), Represent(表示), Epicenter (震央), Earthquake (地震),

Surface (表面), Origin (起源), Seismograph (地震儀), Intersection Point (交點).

Sentence:

1. Zone 1 serves people living within 3 miles of the city’s center. (第一區服務距離市中心 3 英

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7

里以內的人民。)

2. Graph the circles on a coordinate plane. (將圓畫在坐標平面上。) 3. Each unit corresponds to one mile. (每單位為 1 英里。)

參考資料 References

1. 許志農、黃森山、陳清風、廖森游、董涵冬（2019）。數學 1：單元 6 圓方程式。龍騰文化。

2. Big ideas math (2022). Circles in the Coordinate Plane. https://reurl.cc/qNm0Kg.

製作者：臺北市立陽明高中吳柏萱教師