เรขาคณิตวิเคราะห์และ ภาคตัดกรวย ·...

เรขาคณตวเคราะหและภาคตดกรวย อ.ภานวฒน เกยรตนฤมล

1 Think before you will speak.| จงคดกอนททานจะพด (แตจงอยาพดทกค าทคด)

เรขาคณตวเคราะหและภาคตดกรวย

ความรพนฐานเรขาคณตวเคราะห

1. ระยะทางระหวางจด 2 จด

ตวอยาง ก ำหนดพกดจด P(1,-2) และ Q(-5,4) จงหำ PQ

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ตวอยาง ก ำหนดพกดจด A(0,0) และ Q(-1,-2) จงหำ AB

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2. จดกงกลางระหวางจดสองจด

จดก งกลำง =

2,

2

2121 yyxx

ตวอยาง ก ำหนดจดก งกลำงของจดตอไปน

1. P(0,0) และ Q(2,2)

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2. A(-1,-3) และ Q(-2,0) ..........................................................................................................................................................................

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3. A(-1,-3) และ Q(1,1) ..........................................................................................................................................................................

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ตวอยาง ถำ C(2,3) เปนจดก งกลำงระหวำงจด A(-2,0) และ B(x,y) จงหำ B(x,y)

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3. ความชน (slope; m) ของเสนตรง

ตวอยาง จงหำควำมชนทลำกผำนจดในแตละขอตอไปน

1. A(2,3) และ Q(-4,3) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... 2. A(3,2) และ Q(3,-4) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………...

3. A(2,3) และ Q(6,5) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... 4. A(3,2) และ Q(-3,6) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………...



ตวอยาง จงหำคำ x ทท ำใหเสนตรงทผำนจด P และ Q มควำมชนเทำกบ m ตำมทก ำหนดให

1. )3,6( P และ ),9( xQ ; 3

2m

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2. )12,(xP และ )12,5(Q ; 0m ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………...

แบบฝกหด 1.1

1. จงหำระยะหำงระหวำงจดตอไปน 1.1 A(0,0) และ B(-3,4) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... 1.2 A(1,-2) และ B(0,0) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... 1.3 A(-5,0) และ B(4,0) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………...

1.4 P(-2,4) และ Q(5,7) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... 1.5 P(1,2) และ Q(4,6) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... 1.6 P(-4,-5) และ Q(8,-3) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………...



2. จงหำจดก งกลำงระหวำงจดแตละคตอไปน 2.1 A(0,0) และ B(6,8) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... 2.2 A(1,2) และ B(7,6) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... 2.3 A(-3,4) และ B(5,-6) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………...

2.4 A(1,2) และ B(-2,8) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... 2.5 A(-2,-2) และ B(-4,-4) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... 2.6 A(0,2) และ B(2,0) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………...

3. ถำ C(1,2) เปนจดก งกลำงระหวำงจด A(-1,0) และ B(x,y) จงหำ B(x,y)

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5. จงหำควำมชนทลำกผำนจดในแตละขอตอไปน 5.1 A(2,3) และ Q(1,3) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... 5.2 A(1,2) และ Q(1,-4) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... 5.3 A(3,2) และ Q(1,-4) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………...

5.4 A(-3,-3) และ Q(-1,3) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... 5.5 A(-4,2) และ Q(-1,-4) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... 5.6 A(0,2) และ Q(1,0) ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………... ……………………………………………………...

6. จงหำคำ x ทท ำใหเสนตรงทผำนจด P และ Q มควำมชนเทำกบ m ตำมทก ำหนดให

)2,4( P และ ),3( xQ ; 2

1m

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4. สมการของเสนตรง

ตวอยาง จงหำควำมชน และจดตดแกน x จดตดแกน y

1. 032 yx

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2. 1022 yx

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3. 4x

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สมกำรเสนตรงทมควำมชน m ผำนจด (x1, y1) คอ

)( 11 xxmyy

ตวอยาง 1. จงหำสมกำรเสนตรงทผำนจด (1,-2) และมควำมชน 2

1

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2. จงหำสมกำรเสนตรงทผำนจด (-1,2) และมควำมชน 3

2

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จงหำควำมชน และจดตดแกน x จดตดแกน y

1. 052 yx

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2. 102 yx

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3. 10y

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4. จงหำสมกำรเสนตรงทผำนจด (3,-2) และมควำมชน 2

1

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5. จงหำสมกำรเสนตรงทผำนจด (-2,-2) และมควำมชน 4

1

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5. ระยะหางระหวางจดกบเสนตรงและ

ระยะหางระหวางเสนตรงคขนานทงสองเสน

ระยะหำงระหวำงจดกบเสนตรง

ระยะหำงระหวำงเสนตรงคขนำนทงสองเสน



ตวอยาง จงหำระยะหำงระหวำงเสนตรงกบจดตอไปน

1. 0243 yx กบ (2,-3)

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2. 543 yx กบ (0,0)

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จงหำระยะหำงระหวำงเสนตรง 0586 yx กบเสนตรง 01043 yx

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จงหำระยะหำงระหวำงเสนตรง 0152015 yx กบเสนตรง 01043 yx

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จงหำระยะหำงระหวำงเสนตรงกบจดตอไปน

1. 0543 yx กบ (3,4)

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2. 0834 yx กบ (1,2)

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3. 0786 yx กบ (1,-1)

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4. จงหำระยะหำงระหวำงเสนตรง 0543 yx กบเสนตรง 01043 yx

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5. จงหำระยะหำงระหวำงเสนตรง 01034 yx กบเสนตรง 0968 yx

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6. ภาคตดกรวย (Conic Section)



7. วงกลม (circle)



ตวอยาง จงหำสมกำรวงกลม ก ำหนดให

1. จดศนยกลำงอยท (0,0) และรศมยำว 2 หนวย

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3. จงหำรศมและจดศนยกลำงของสมกำรวงกลม

3.1 4)3()2( 22 yx

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3.2 10)3()2( 22 yx

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3.3 16)1( 22 yx

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3.4 20)1( 22 yx

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4.1 034622 yxyx

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4.2 069241022 yxyx

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1. จงหำสมกำรวงกลม ก ำหนดให


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4. จดศนยกลำงอยท (-1,-2) และรศมยำว 2 หนวย

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5. จดศนยกลำงอยท (-4,0) และรศมยำว 3

2 หนวย

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1. 49)1( 22 yx …………………………………………………… …………………………………………………… …………………………………………………… …………………………………………………… …………………………………………………… …………………………………………………… ……………………………………………………

2. 16)1()3( 22 yx …………………………………………………… …………………………………………………… …………………………………………………… …………………………………………………… …………………………………………………… …………………………………………………… ……………………………………………………



3. 25)2( 22 yx …………………………………………………… …………………………………………………… …………………………………………………… …………………………………………………… …………………………………………………… …………………………………………………… ……………………………………………………

4. 9)2()1( 22 yx …………………………………………………… …………………………………………………… …………………………………………………… …………………………………………………… …………………………………………………… …………………………………………………… ……………………………………………………


3.1 07622 xyx

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3.2 0510822 yxyx

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โจทยวดความสามารถในการแกปญหาจากสถานการณจรง เรองวงกลม

เดกเลยงไก 2 คนวำงงำนมำกจงพำกนมองบนทองฟำเหนดำว “ลกไกตวเดยว” โคจรรอบดำว แมไก เปนวงกลมมสมกำรเคลอนทคอ 4)3()2( 22 yx เสมอตำมกำรค ำนวณของ เดกเลยงไก คนแรก อยมำวนหนงเดกเลยงไกอกคนกเหนมดำว “ลกเปดหลงทำง” เคลอนท ลอมรอบดำว “พอไกตำงเลำ” เปนวงกลมเชนกนและมสมกำรกำรเคลอนทของดำวลกเปดหลงทำงคอ 014222 yxyx เสมอ ถำหำกทงสองวงโคจรอยบนระนำบเดยวกนตำม กำรค ำนวณของเดกเลยงไกทงสองและดำวแมไกกบดำวพอไกตำงเลำไมเคลอนทเลย เดกเลยงไกทงสองก ำลงเถยงกนเก ยวกบวงโคโจรของดำวลกไกตวเดยว และดำวลกเปดหลงทำงโดย คนแรกบอกวำ “วงโคจรของดำวตดกน แสดงวำ อำจจะเกดกำรชนกนของดำวทงสองนได” คนทสองบอกวำ “ไมมทำงทจะชนกน เพรำะวงโคจรของดำวทงสองไมตดกนเลย” ใหนกเรยนวเครำะหวำค ำพดของใครถกตอง พรอมใหเหตผลประกอบ …......................................................................................................................................................................

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8. พาราโบลา (Parabola)



ตวอยาง จงหำสมกำรพำรำโบลำ จำกสงทก ำหนดให (พรอมทงวำดกรำฟประกอบ)

1. จดยอดอยท (-2,3) และจดโฟกสอยท (1,3)

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2. จดยอดอยท (-3,-4) และเสนไดเรกตรกซคอเสนตรง x=2

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3. ไดเรกตรกซคอเสนตรง x=5 และจดโฟกสอยท (2,4)

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4. จงหำจดยอด โฟกส ไดเรกตกซ แกนพำรำโบลำ ควำมยำวของลำตสเรกตม พรอมทงเขยนกรำฟ จำก

สมกำรพำรำโบลำแตละขอตอไปน

4.1 xy 82 4.2 yx 42

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4.3 0562042 yxx

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4.4 019622 xyy

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จงหำสมกำรพำรำโบลำ จำกสงทก ำหนดให (พรอมทงวำดกรำฟประกอบ)

1. จดยอดอยท (0,0) และจดโฟกสอยท (6,0)

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2. จดยอดอยท (0,0) และเสนไดเรกตรกซคอเสนตรง x=3

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3. ไดเรกตรกซคอเสนตรง x=3 และจดโฟกสอยท (-3,0)

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4. ไดเรกตรกซคอเสนตรง y=4 และจดโฟกสอยท (0,-4)

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จงหำจดยอด โฟกส ไดเรกตกซ แกนพำรำโบลำ ควำมยำวของลำตสเรกตม พรอมทงเขยนกรำฟ จำกสมกำร

พำรำโบลำแตละขอตอไปน

4.1 xy 42 4.2 yx 162

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4.3 0331622 xyy

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4.4 01862 yxx

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โจทยวดความสามารถในการแกปญหาจากสถานการณจรง เรองพาราโบลา

กำรสลำยนวเปนกรรมวธรกษำ

โรคนวในทำงเดนปสสำวะทไมตองผำตด

ไมมบำดแผล และไมตองดมยำสลบ ท ำ

โดยกำรใชคลนเสยงทมควำมจ ำเพำะ ท ำ

ใหเกดแรงกระแทกทกอนนว ในลกษณะ

ของรปพำรำโบลำท ำใหเกดรอยรำวจน

แตกเปนผง ผงนวจะหลดไหลออกมำ

พรอมกบน ำปสสำวะ

คลนเสยงทก ำเนดมำจำกเครองสลำยนวจะถกควบคมใหพงสกอนนวโดยไมท ำอนตรำยกบเนอเยอ

ของผปวย ก ำหนดใหกอนนวอยทจด (0 , 0) คำ 1c (ระยะหำงระหวำงจดยอด และจดโฟกส) และ

ก ำหนดจด (h , k) ทจด (15 , 0) , (10 , 0) , (5 , 0) , (0 , 0) ตำมล ำดบ ทต ำเหนงของเครองสลำยนว จงหำ

สมกำรของคลนเสยงทถกยงมำยงกอนนว

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9. วงร (Ellipse)

ภำพทเหนในชวตประจ ำวน



ตวอยาง จำกสมกำรวงรในแตละขอตอไปน จงหำจดศนยกลำง จดยอด โฟกส จดปลำยแกนโท ควำมยำว

ลำตสเรกตม (พรอมทงวำดกรำฟประกอบ)

1. 1916

22

yx

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2. 194

22

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3. 33 22 yx

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ตวอยาง จำกสงทก ำหนดใหตอไปน จงหำสมกำรวงร (วำดรปประกอบ)

1. โฟกสอยทจด (3,0) และ (-3,0) และผลบวกของระยะจำกจดใดๆไปยงโฟกสทงสอง (ผลบวกคำคง

ตว) เทำกบ 8 หนวย

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2. จดยอดอยทจด (5,0) และ (-5,0) โฟกสจดหนงอยท (2,0)

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1. จำกสมกำรวงรในแตละขอตอไปน จงหำจดศนยกลำง จดยอด โฟกส จดปลำยแกนโท ควำมยำวลำตสเรก

ตม (พรอมทงวำดกรำฟประกอบ)

1. 11625

22

yx

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2. 1259

22

yx

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3. 3694 22 yx

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2. จำกสงทก ำหนดใหตอไปน จงหำสมกำรวงร (วำดรปประกอบ)

1. โฟกสอยทจด (-3,0) และ (3,0) และผลบวกของระยะจำกจดใดๆไปยงโฟกสทงสอง (ผลบวกคำคง

ตว) เทำกบ 8 หนวย

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2. จดยอดอยทจด (0,5) และ (0,-5) โฟกสจดหนงอยท (0,2)

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3. จดยอดอยท (-10,0) และ (10,0) และโฟกสอยท (-8,0) และ (8,0)

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โจทยวดความสามารถในการแกปญหาจากสถานการณจรง เรองวงร

สนำมหญำดำนหลงท ำเนยบขำวในกรงวอชงตน

ด ซ ของสหรฐอเมรกำ เปนรปวงรมควำม ยำว 616 ฟต

กวำง 528 ฟต รอบๆบรเวณนนเปนทตงของสวนธรกจ

และทท ำกำรไปรษณยเกำแกของ เมองซงสำมำรถ

มองเหนไดจำกบนเขำวอชงตน จงเขยนสมกำรวงรของ

สนำม

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10. ไฮเพอรโบลา (Hyberbola)



ตวอยาง จำกสมกำรไฮเพอรโบลำในแตละขอตอไปน จงหำจดศนยกลำง จดยอด โฟกส จดปลำยแกนสงยค

สมกำรเสนก ำกบ ควำมยำวลำตสเรกตม eccentricity หรอ e (พรอมทงวำดกรำฟประกอบ)

1. 1416

22

xy

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2. 1416

22

yx

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3. 3694 22 yx

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จำกสงทก ำหนดใหตอไปน จงหำสมกำรไฮเพอรโบลำ (วำดรปประกอบ)

1. ผลตำงของระยะจำกจดใดๆบนไฮเพอรโบลำไปยงจด (4,2) และ (-2,2) เทำกบ 4

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2. โฟกสอยทจด (-2,-6) และ (-2,4) และผลตำงคงตวเทำกบ 6 หนวย

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โจทยวดความสามารถในการแกปญหาจากสถานการณจรง เรองไฮเพอรโบลา

วงโคจรของดำวหำงทมควำมเรวสงบำงดวง

โดยเฉพำะดำวหำงแบบปรำกฏครงเดยว (Single-apparition

comets) ซงจะผำนเขำมำในระบบสรยะเพยงครงเดยวนน

จะมวถโคจรเปนรปไฮเพอรโบลำ โดยมดวงอำทตยเปน

โฟกสจดหนง ถำดำวหำงดวงหนงมวถโคจรแสดงไดดวย

สมกำร 400,846400116,2 22 yx ต ำแหนงทดำวหำง

อยใกล ดวงอำทตยมำกทสดเปนเทำใด ถำก ำหนดให

ระยะทำงมำหนวยเปนลำนไมล

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1. จำกสมกำรไฮเพอรโบลำในแตละขอตอไปน จงหำจดศนยกลำง จดยอด โฟกส จดปลำยแกนสงยคสมกำร

เสนก ำกบ ควำมยำวลำตสเรกตม eccentricity หรอ e (พรอมทงวำดกรำฟประกอบ)

1. 12516

22

xy

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2. 12516

22

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3. 1004 22 yx

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2. จำกสงทก ำหนดใหตอไปน จงหำสมกำรไฮเพอรโบลำ (วำดรปประกอบ)

2.1 ผลตำงของระยะจำกจดใดๆบนไฮเพอรโบลำไปยงจด (-3,2) และ (3,2) เทำกบ 4

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2.2 โฟกสอยทจด (-2,-6) และ (-2,4) และผลตำงคงตวเทำกบ 4 หนวย

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ตารางการสงงานเรองเรขาคณตวเคราะหและภาคตดกรวย

วน/เดอน/ป งำน ลำยเซนผปกครอง ลำยเซนคร แบบฝกหดท 1.1 แบบฝกหดท 1.2 แบบฝกหดท 1.3 แบบฝกหดท 1.4 แบบฝกหดท 1.5 แบบฝกหดท 1.6 แบบฝกหดท 1.7 โจทยวดควำมสำมำรถในกำรแกปญหำ

จำกสถำนกำรณจรง เรองวงกลม

โจทยวดควำมสำมำรถในกำรแกปญหำจำกสถำนกำรณจรง เรองพำรำโบลำ

โจทยวดควำมสำมำรถในกำรแกปญหำจำกสถำนกำรณจรง เรองไฮเพอรโบลำ

โจทยวดควำมสำมำรถในกำรแกปญหำจำกสถำนกำรณจรง เรองวงร



ผงความคด (Mind Mapping) เรองเรขาคณตวเคราะหและภาคตดกรวย

เรขาคณตวเคราะหและภาคตดกรวย

http://www.si.mahidol.ac.th/th/division/soqd/admin/news_files/403_18_4.pdf



แบบทดสอบกอน-หลงเรยน เรองเรขาคณตวเคราะหและภาคตดกรวย

ใหนกเรยนท าเครองหมายกากบาท ( ) ลงในชอง ก, ข, ค หรอ ง ทนกเรยนเหนวาถกตองทสดเพยงค าตอบเดยว

1. ขอใดคอระยะหางระหวางจด A(2, 1) และ

จด B(2, -2)

ก. 1 หนวย

ข. 3 หนวย

ค. 5 หนวย

ง. 9 หนวย

2. จด B(4, 3) อยหางจากแกน Y กหนวย

ก. 4 หนวย

ข. 6 หนวย

ค. 8 หนวย

ง. 10 หนวย

3. ขอใดคอจดกงกลางระหวางจด A(0, 0) และ

B(6, 8)

ก. (0, 3)

ข. (3. 8)

ค. (2, 4)

ง. (3, 4)

4. ขอใดคอความชนของเสนตรงทผานจด

A(-2, -6) และ B(3, -1)

ก. 1

ข. 2

ค. -1

ง. -2

5. เสนตรงทผานจดสองจดใด

มความเทากบ 158

ก. (3, 8 ) และ ( 5, 7 )

ข. (3, 8 ) และ ( 6 9 )

ค. (2, 8 ) และ ( 5, 7 )

ง. ( 5, 8 ) และ ( 5, 9 )

6. ขอใดคอระยะหางระหวางเสนตรง

3x + 4y +5 = 0 กบจด A(3, 4)

ก. 5 หนวย

ข. 6 หนวย

ค. 7 หนวย

ง. 8 หนวย



7. ระยะหางระหวางเสนคขนานทมสมการ

4x - 3y - 10 = 0 และ 8x – 6y + 9 = 0

มคาเทาใด

ก. 115

หนวย

ข. 295

หนวย

ค. 115

หนวย

ง. 295

หนวย

8. ขอใดคอสมการวงกลมทมจดศนยกลาง

(13

,23

) และรศมยาว 2 หนวย

ก. 2 21 2(x ) (y ) 2

3 3

ข. 2 2 21 2(x ) (y ) 2

3 3

ค. 2 2 21 2(x ) (y ) ( 2 )

3 3

ง. 2 2 21 2(x ) (y ) ( 2 )

3 3

9. ขอใดคอสมการพาราโบลาทมจดยอดอยท

(0, 0) และจดโฟกสอยท (6, 0)

ก. 2y 6x

ข. 2y 24x

ค. 2x 12y

ง. 2x 24y

10. ก าหนดสมการวงร

2 2(x 2) (y 1)

1100 25

จดยอดคอขอใด

ก. ( 8,1) และ (12,1)

ข. (8,1) และ (12,1)

ค. (8,10) และ ( 10,1)

ง. (10,5) และ (5,10)

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