Topic: the locus of a hyperbola

Topic: the locus of a hyperbola

1. Watch this clip to 1:49. Share with your partner what you see in this clip.

2. Check these words

3. Watch this clip from 1:49. Then finish the following geometric definiBon of the hyperbola.

條件：_____________________________________________________

規則： P is a point …..

定義：

A hyperbola is the set of all points (x,y)…..

English !" #$

Hyperbola Hyperboloid Hyperbolic

4. 雙曲線的名詞要素：

(1) Focus/Foci： (2) Transverse axis：

(3) Center：

(4) Vertex/verCces：

(5) Length of the transverse axis:

(6) The distance between the foci:

(7) The distance from center to either focus:

(8) Conjugate axis:

(9) Co-verCces:

(10) Length of the conjugate axis:

(11) Asymptote/asymptotes:

5. Property of a hyperbola： If a hyperbola with transverse and conjugate axes of lengths 2a and 2b respecCvely. The distance between the foci is 2c, then _____________.

6. The diagram below shows several concentric circles centered at points and . The radius of each circle is one unit away from the adjacent circle. Use the definiCon of a hyperbola to sketch it.

(1) The moving point P on the hyperbola saCsfy

(2) Find the length of the transverse axis、 the distance between the foci and the length of the conjugate axis.

F₁ F₂

|PF₁− PF₂| = 6

(3) Find a、b、c.

7.如圖，在方格紙中有兩組同心圓，圓心分別為 F1 與 F2，若 P 點在以 F1，F2 為焦點的雙曲線 上，判斷下列各點是否在此雙曲線上。

( ) A ( ) B ( ) C ( ) D ( ) E

8.如圖，設P為雙曲線的一點且 ，若F1，F2為此雙曲線的兩焦點且 ，已

知 ，則△F¹PF2的周長為多少？

9. Challenge：

(1)( 2a=10,2c=10 ) The set of all points P(x,y) that saBsfies . Determine what kind of graph will we get from using the concentric circle centered at points and .

|PF₁− PF₂| = 6 F₁F₂= 10

PF₁: PF₂ = 1 : 3

|PF₁− PF₂| = 10 F₁ F₂

(2) ( 2a=20,2c=10 ) The set of all points P(x,y) that saCsfies . Determine what kind of graph will we get from using the concentric circle centered at points and .

(3) ( 2a=0,2c=10 ) The set of all points P(x,y) that saCsfies . Determine what kind of graph will we get from using the concentric circle centered at points and .

|PF₁− PF₂| = 20 F₁ F₂

|PF₁− PF₂| = 0 F₁ F₂

Conclusion：

(1) If 2a<2c, the set of all points P(x,y) that saCsfies is a hyperbola.

(2) If 2a＝2c, the set of all points P(x,y) that saCsfies is two rays (two half-lines).

(3) If 2a>2c, the set of all points P(x,y) that saCsfies is an empty set.

(4) If 2a=0, the set of all points P(x,y) that saCsfies is the perpendicular bisector of .

|PF₁− PF₂| = 2a

|PF₁− PF₂| = 2a

|PF₁− PF₂| = 2a

|PF₁− PF₂| = 2a F₁F₂

Topic: the locus of a hyperbola使⽤建議

1. Watch this clip to 1:49. Share with your partner what you see in this clip.

[教學活動安排]

讓學⽣透過觀看影片引發學習雙曲線的動機,並可利⽤此機會練習英聽及筆記擷取重點

[可參考的英⽂問句/提問/開場]

In this class, we are going to learn about hyperbola. What is a hyperbola and why should we learn about it?

Let’s watch this clip to 1:49. Then, share with your partner what you see in this clip.

2. Check these words [教學活動安排]

教師可讓學⽣兩兩⼀組,⽤⼿機上網查這些將會使⽤到的單字、關鍵字及發⾳並完成表格 也可以請學⽣將在這堂課中遇到不熟的單字記錄下來

3. Watch this clip from 1:49. Then finish the following geometric definiBon of the hyperbola.

條件：Two points on the plane and . The difference of distance from and is constant( )

規則：P is a point saCsfied

English !" #$

Hyperbola Hyperboloid Hyperbolic

F₁ F₂ F₁

F₂ α

|PF₁− PF₂| = α

定義：

A hyperbola is the set of all points (x,y) in a plane, the difference of whose distances from two disCnct fixed points( and ) are constant

[教學活動安排]

Think-pair-share

利⽤全英⽂影片中的定義說明，讓學⽣練習在全英的語境下理解雙曲線的幾何定義，有需要時 可以讓學⽣多聽幾次。當學⽣完成時，讓學⽣兩兩⼀組，互相跟對⽅說⾃⼰從影片中聽到並寫下 來對於雙曲線的幾何定義。

註：紅字部分為參考解答

4. 雙曲線的名詞要素：

(1) Focus/Foci：

(2) Vertex/verCces：

(3) Transverse axis：

(4) Center：

(5) Length of the transverse axis:

(6) The distance between the foci:

(7) The distance from center to either focus:

(8) Conjugate axis:

(9) Co-verCces:

(10) Length of the conjugate axis:

(11) Asymptote/asymptotes:

F₁ F₂

[教學活動安排]

教師介紹並說明

[可參考的英⽂問句/提問/說明]

The graph of a hyperbola has two disconnected parts(branches). How many axes of symmetry does the hyperbola have? Can you draw it?

(1) Focus/Foci： The two fixed points 、 are called foci (focus是單數，foci是複數).

(2) Vertex/verCces： The line through the foci intersects the hyperbola at two points are called verCces. We denote them as A and B.

(3) Transverse axis（貫軸）： The line segment connecCng verCces A and B is the transverse axis.

(4) Center： The midpoint of the transverse axis.

(5) Length of the transverse axis:

By the definiCon, we know that and (symmetry) Therefore,

(6) The distance between the foci: . We denote it as 2c (7) The distance from center to either focus: c

(8) Conjugate axis（共軛軸）: The line segment with length 2b through the center and

perpendicular to the transverse axis is called the conjugate axis. 與貫軸垂直於中⼼且長度為2b的線 段

(9) Co-verCces(共軛軸頂點): The endpoints of the conjugate axis. We denote them as C and D.

(10) Asymptote/asymptotes(漸近線): Every hyperbola has two asymptotes that intersect at the center of the hyperbola, as shown in Figure. The asymptotes pass through the verCces of a rectangle of the dimensions 2a by 2b.

5. Property of a hyperbola： If a hyperbola with transverse and conjugate axes of lengths 2a and 2b respecCvely. The distance between the foci is 2c, then _____________.

[教學活動安排]

強調雙曲線中a、b、c的關係,

F₁ F₂

AB = 2a

AF₂− AF₁= 2a AF₁= BF₂ 2a = AF₂− AF₁ = AF₂− BF₂= AB

F₁F₂ = 2c

c²= a²+ b²

6. The diagram below shows several concentric circles centered at points and . The radius of each circle is one unit away from the adjacent circle. Use the definiCon of a hyperbola to sketch it.

(1) The moving point P on the hyperbola saCsfy

(2) Find the length of the transverse axis、 the distance between the foci and the length of the conjugate axis.

(3) Find a、b、c.

Answer: (1)

(2) 6、10、8 (3) 3、5、4

7.如圖，在方格紙中有兩組同心圓，圓心分別為 F1 與 F2，若 P 點在以 F1，F2 為焦點的雙曲線 F₁ F₂

|PF₁− PF₂| = 6

上，判斷下列各點是否在此雙曲線上。

( ) A ( ) B ( ) C ( ) D ( ) E

Answer: ( X ) A ( X ) B ( O ) C ( O ) D ( X ) E

8.如圖，設P為雙曲線的一點且 ，若F1，F2為此雙曲線的兩焦點且 ，已

知 ，則△F1PF2的周長為多少？

Answer: 22

9. Challenge：

(1)( 2a=10,2c=10 ) The set of all points P(x,y) that saBsfies . Determine what kind of graph will we get from using the concentric circle centered at points and .

Answer:

|PF₁− PF₂| = 6 F₁F₂= 10

PF₁: PF₂ = 1 : 3

|PF₁− PF₂| = 10 F₁ F₂

(2) ( 2a=20,2c=10 )The set of all points P(x,y) that saCsfies . Determine what kind of graph will we get from using the concentric circle centered at points and .

Answer: no graph（無圖形）.

|PF₁− PF₂| = 20 F₁ F₂

(3) ( 2a=0,2c=10 ) The set of all points P(x,y) that saCsfies . Determine what kind of graph will we get from using the concentric circle centered at points and .

Answer:

Conclusion：

(1) If 2a<2c, the set of all points P(x,y) that saCsfies is a hyperbola.

(2) If 2a＝2c, the set of all points P(x,y) that saCsfies is two rays (two half-lines).

(3) If 2a>2c, the set of all points P(x,y) that saCsfies is an empty set.

(4) If 2a=0, the set of all points P(x,y) that saCsfies is the perpendicular bisector of .

|PF₁− PF₂| = 0 F₁ F₂

|PF₁− PF₂| = 2a

|PF₁− PF₂| = 2a

|PF₁− PF₂| = 2a

|PF₁− PF₂| = 2a F₁F₂

製作者：台北市立育成⾼中 林⽟惇