Topic: the locus of an ellipse
1. Watch this clip to 2:16. Share with your partner what you see in this clip.
2. Check these words
3. Watch this clip from 2:16. Then finish the following geometric definiAon of the ellipse.
條件：_____________________________________________________
規則： P is a point …..
定義：
An ellipse is the set of all points (x,y)…..
English !" #$
Ellipse Foci
4. 橢圓的名詞要素：
(1) Focus/Foci： (2) Major axis：
(3) Minor axis：
(4) Center：
(5) Vertex/verAces：
(6) Co-verAces :
(7) Length of the major axis:
(8) Length of the minor axis:
(9) The distance between the foci :
(10) The distance from center to either focus:
3. Property of an ellipse： If an ellipse with major and minor axes of lengths 2a and 2b, respecAvely, where 0<b<a. The distance between the foci is 2c, then _____________.
4. 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 definiAon of an ellipse to sketch it.
(1) The moving point P on the ellipse saAsfy
(2) Find the length of the major axis、 the distance between the foci and the length of the minor axis (3) Find a、b、c
F1 F2 PF1+ PF2= 16
5.如圖，以O(0,0)為圓⼼，半徑為1﹐2﹐3畫三個同⼼圓，以P(4,0)為圓⼼，半徑為1﹐2﹐3﹐4畫四 個同⼼圓。若A﹐B﹐C﹐D﹐E﹐F在某⼀個橢圓上，則關於該橢圓，下列哪些選項是正確的﹖
(1)中⼼為(2,0) (2)長軸長為4 (3)短軸長為3 (4)(4,0)為其焦點
6.⼀橢圓形撞球台，其長軸長為10，且其兩焦點為F1，F2，今有⼀⼈從F2 將⼀球依直線⽅向打⾄
邊上⼀點P，反彈過F1，撞⾄邊上另⼀點Q，再回到原焦點處F2，試求 ΔPQF2 的周長。
7. Challenge：
(1)( 2a=10，2c=10 )， The set of all points P(x,y) saAsfies . Determine what kind of graph will we get from using the concentric circle centered at points F1 and F2.
(2) ( 2a=4，2c=10 )， The set of all points P(x,y) saAsfies . Determine what kind of graph will we get from using the concentric circle centered at points F1 and F2.
Conclusion：(1) If 2a＞2c， the set of all points P(x,y) saAsfies is an ellipse.
(2) If 2a＝2c， the set of all points P(x,y) saAsfies is a line segment . (3) If 2a＜2c， the set of all points P(x,y) saAsfies is an empty set.
PF1+ PF2= 10
PF1+ PF2= 4
PF1+ PF2= 2a
PF1+ PF2= 2a F1F2
PF1+ PF2= 2a
Topic: the locus of an ellipse使⽤建議
1. Watch this clip to 2:16. 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橢圓的動機 並可利⽤此機會練習英聽及記筆記擷取重點 [可參考的英⽂問句/提問/開場]
In this class, we are going to learn about the ellipse. What is an ellipse and why should we learn about it?
Let’s watch this clip to 2:16. 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的不熟的單字記錄下來
3. Watch this clip from 2:16.
Then finish the following geometric definiAon of the ellipse.
條件： Two points on the plane 、 . The Sum of distance from and is constant( ).
規則： P is a point that 定義：
An ellipse is the set of all points (x,y) in a plane, the sum of whose distances from two disAnct fixed points( 、 ) is constant.
English 中⽂ 圖⽰
Ellipse Foci
F1 F2 F1 F2 α
PF1+ PF2= α
F1 F2
[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橢圓的幾何定義
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 並寫下來對於橢圓的幾何定義．
註：紅字部分為參考解答 4. 橢圓的名詞要素：
(1) Focus/Foci：
(2) Major axis：
(3) Minor axis：
(4) Center：
(5) Vertex/verAces：
(6) Co-verAces :
(7) Length of the major axis:
(8) Length of the minor axis:
(9) The distance between the foci :
(10) The distance from center to either focus:
[教學活動安排]
教師介紹並說明
可參考這影片hcps://youtu.be/Uy-Ig5zgNsE
(1) Focus/Foci： The two fixed points are called the foci (plural of focus).
(2) Major axis： The line segment through the foci is called the major axis.
(3) Minor axis： The line segment through the center and perpendicular to the major axis is called the minor axis.
(4) Center： The mid-point of the major axis.
(5) Vertex/verAces： The endpoints of the major axis. We denote them as A and B (6) Co-verAces: The endpoints of the minor axis. We denote them as C and D
(7) Length of the major axis:
By the definiAon, we know that and (symmetry) Therefore,
(8) Length of the minor axis: (We denote it as 2b, we will explain this later when introducing the property)
(9) The distance between the foci: (We denote it as 2c) (10) The distance from center to either focus: c
3. The property of the ellipse： If an ellipse with major and minor axes of lengths 2a and 2b, respecAvely, where 0<b<a. The distance between the foci is 2c, then ________.
[教學活動安排]
A B = 2a
AF1+ AF2= 2a BF2= AF1
2a = AF1+ AF2= BF2+ AF2= A B CD = 2b
F1F2= 2c
強調橢圓中a、b、c的關係,
4. 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 definiAon of an ellipse to sketch it.
(1) The moving point P on the ellipse saAsfy
(2) Find the length of the major axis、 the distance between the foci and the length of the minor axis (3) Find a、b、c
[教學活動安排]
在此之前都是介紹橢圓的幾何定義、性質,因此接下來利⽤同⼼圓及例⼦讓學⽣實際練習⽤幾 何定義去畫出橢圓，同時透過此活動驗收學⽣對橢圓幾何定義是否掌握
Answers:
(1)
a2 = b2+ c2
F1 F2 PF1+ PF2= 16
(2) 16、10、
(3) 8、5、
5.如圖，以O(0,0)為圓⼼，半徑為1﹐2﹐3畫三個同⼼圓，以P(4,0)為圓⼼，半徑為1﹐2﹐3﹐4畫四
個同⼼圓。若A﹐B﹐C﹐D﹐E﹐F在某⼀個橢圓上，則關於該橢圓，下列哪些選項是正確的﹖
(1)中⼼為(2,0) (2)長軸長為4 (3)短軸長為3 (4)(4,0)為其焦點
[%&'()*]
+,-./&0123456789:;<=>?@A Answer:(1)(3)(4)
6.⼀橢圓形撞球台，其長軸長為10，且其兩焦點為F1，F2，今有⼀⼈從F2 將⼀球依直線⽅向打⾄
邊上⼀點P，反彈過F1，撞⾄邊上另⼀點Q，再回到原焦點處F2，試求 ΔPQF2 的周長。
[%&'()*]
+,-./&0123456789:;<=>?@A Answer:20
7. Challenge：
(1)( 2a=10，2 c=10 )， The set of all points P(x,y) saAsfies . Determine what kind of graph will we get from using the concentric circle centered at points and .
2 39 39
PF1+ PF2= 10 F1 F2
(2) ( 2a=4，2 c=10 )， The set of all points P(x,y) saAsfies . Determine what kind of graph will we get from using the concentric diagram centered at points and
Conclusion：(1) If 2a＞2c， the set of all points P(x,y) saAsfies is an ellipse.
(2) If 2a＝2c， the set of all points P(x,y) saAsfies is a line segment . (3) If 2a＜2c， the set of all points P(x,y) saAsfies is an empty set, there is no graph.
PF1+ PF2= 4
F1 F2
PF1+ PF2= 2a
PF1+ PF2= 2a F1F2
PF1+ PF2= 2a
[教學活動安排]
此處教師可⾃⾏依學⽣程度安排是否要加入教學活動中 Answer:
(1)
(2)We can’t find any points. So there is no graph.
參考資料：
1. hcps://youtu.be/nzwCInIMlU4 2. hcps://youtu.be/Uy-Ig5zgNsE
3. hcps://www.slideshare.net/itutor/ellipse-26682425
製作者：台北市立育成⾼中 林⽟惇老師
