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แบบฝึ�กทั�กษะคณิ�ตศาสตร์�เพิ่��มเต�ม ร์หั�ส ค 31202
เร์��อง เร์ขาคณิ�ตวิ�เคร์าะหั�
ภาคเร์�ยนทั�� 2 ปี#การ์ศ$กษา ……………..
โร์งเร์�ยนกาญจนาภ�เษกวิ�ทัยาลั�ย เพิ่ชร์บ*ร์ณิ�
ช��อ – สก+
ลั.......................................................................................เลัขทั��......................ม. 4/……………………..
โร์งเร์�ยนกาญจนาภ�เษกวิ�ทัยาลั�ย เพิ่ชร์บ*ร์ณิ� ส,าน�กงานเขตพิ่�-นทั��การ์ศ$กษาม�ธยมศ$กษา เขต 40
ส,าน�กงานคณิะกร์ร์มการ์การ์ศ$กษาข�-นพิ่�-นฐาน กร์ะทัร์วิงศ$กษาธ�การ์
แบบฝึ�กหั�ดทั�� 1
1. จงหาระยะทางระหว่างจ�ดสองจ�ดต่อไปนี้��1.1 (3, 4) และ (0 , 0)
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1.2 (0 , 0) และ (-1, -3)
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1.3 (2, 5) และ (9 , 5)
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1.4 (-5, -6) และ (-5 , -3)
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1.5 (-1, -2) และ (3 , -4)
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1.6 (2, 13) และ (8 , 5)
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1.7 (-5, 3) และ (0 , 8)
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2. ABCD เป�นี้ร�ปส��เหล��ยมคางหม�ด�งร�ป โดยท�� BC ยาว่ 8 หนี้ว่ย ถ้!าจ�ด A ม�พิ#กั�ดเป�นี้ (-4 , 0) จ�ด B ม�พิ#กั�ดเป�นี้ (-1 , 4) และพิ%�นี้ท��ของร�ปส��เหล��ยมคางหม�นี้��เทากั�บ 48
ต่ารางหนี้ว่ย จงหา 2.1 พิ#กั�ดของจ�ด C2.2 คว่ามยาว่ของสว่นี้ของ
เส!นี้ต่รง AB
2.3 พิ#กั�ดของจ�ด D
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Y
X
C
D
B (-1,4)
A (-4 ,0)
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……………………………………………………………………………………………………….3. จงแสดงว่าจ�ด (1 ,1) , (-1,-1) และ (-4,1) เป�นี้จ�ดยอดของร�ปสามเหล��ยมม�มฉากั
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……………………………………………………………………………………………………….4. จงหาระยะหางระหว่างจ�ด (-3 , -4) กั�บแกันี้ X
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………………………………………………………………………………………………………5. จงหาคว่ามยาว่ของเส!นี้รอบร�ปของร�ปสามเหล��ยม ABC ซึ่*�งจ�ด A
ม�พิ#กั�ดเป�นี้ (3 , 4) จ�ด B ม�พิ#กั�ดเป�นี้ (7 , 8) และจ�ด C ม�พิ#กั�ดเป�นี้ (-1 , -2)
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6. จงหาจ�ดซึ่*�งอย�บนี้แกันี้ Y และอย�หางจากัจ�ด (2 , 5) และ (3 ,
-7) เป�นี้ระยะทางเทากั�นี้
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……………………………………………………………………………………………………….7. ร�ปสามเหล��ยมร�ปหนี้*�งม�จ�ด (6 , 8) , (4 , 6) และ (-2 , -2)
เป�นี้จ�ดยอด ร�ปสามเหล��ยมร�ปนี้��เป�นี้ร�ปสามเหล��ยมหนี้!าจ��ว่หร%อไม
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……………………………………………………………………………………………………….8. ว่งกัลมร�ปหนี้*�งม�จ�ดศู�นี้ย,กัลางท��จ�ด (-3 , 2) และผ่านี้จ�ด (7 , 4)
จงหาคว่ามยาว่ของร�ศูม�ของว่งกัลมนี้��
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9. ร�ปสามเหล��ยมร�ปหนี้*�งม�จ�ด (10 , 0) , (-12 , 0) และ (-8 ,
8) เป�นี้จ�ดยอด จงหาพิ%�นี้ท��ของร�ปสามเหล��ยมนี้��
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10. จ�ด (0 , 0) , (8 , 18) และ (12 , 27) อย�บนี้เส!นี้ต่รงเด�ยว่กั�นี้หร%อไม
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……………………………………………………………………………………………………….11. จงบอกัเง%�อนี้ไขท��ท.าให!จ�ด P1 , P2 และ P3 ใดๆ อย�บนี้เส!นี้ต่รงเด�ยว่กั�นี้
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………………………………………………………………………………………………………12. ว่งกัลมว่งหนี้*�งม�จ�ดศู�นี้ย,กัลางท��จ�ด (3 , 4) และผ่านี้จ�ด (6 , 8)
จงหาคว่ามยาว่ร�ศูม�ของว่งกัลมนี้�� และต่รว่จสอบด�ว่าจ�ด (0 , 0) อย�บนี้ว่งกัลมนี้��หร%อไม
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……………………………………………………………………………………………………….13. ว่งกัลมว่งหนี้*�งม�จ�ดศู�นี้ย,กัลางท��จ�ด ( 4 , -3) และม�แกันี้ X เป�นี้เส!นี้ส�มผ่�ส จงหาจ�ดส�มผ่�ส
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แบบงฝึ�กหั�ดทั�� 2
1. จงหาจ�ดกั*�งกัลางระหว่างจ�ดสองจ�ดต่อไปนี้�� 1.1 (12 , 2) และ (3 , -1)
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1.2 (-1 , -3) และ (5 , 3)
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1.3 (3 , -52) และ (-3 , -9)
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1.4 (-3 , -2) และ (-1 , -1)
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1.5 ( 2 , 7) และ ( -2 , 7)
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2. จ�ด M เป�นี้จ�ดกั*�งกัลางของสว่นี้ของเส!นี้ต่รง PQ จงหาพิ#กั�ดของจ�ด P ถ้!า
2.1 M ม�พิ#กั�ดเป�นี้ (1 , 2) และ Q ม�พิ#กั�ดเป�นี้ (3 , 4)
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2.2 M ม�พิ#กั�ดเป�นี้ (5 , 6) และ Q ม�พิ#กั�ดเป�นี้ (15 ,- 4)
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……………………………………………………………………………………………………….3. จงแสดงว่าจ�ดกั*�งกัลางของเส!นี้ทแยงม�มของร�ปส��เหล��ยมซึ่*�งม�จ�ดยอดท�� A (2 , 1) , B (7 , 1) , C (9 ,3) และ D (4 , 3) เป�นี้จ�ดเด�ยว่กั�นี้
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……………………………………………………………………………………………………….4. จงหาคว่ามยาว่ของสว่นี้ของเส!นี้ต่รงซึ่*�งเชื่%�อมจ�ด A (8 , 2) กั�บจ�ดกั*�งกัลางระหว่าง P (2 , 1) และ Q (6 ,5)
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……………………………………………………………………………………………………….5. ถ้!าจ�ดปลายของเส!นี้ผ่านี้ศู�นี้ย,กัลางของว่งกัลมว่งหนี้*�งเป�นี้ (1 , 3)
และ (7 , 11) จงหาพิ#กั�ดของ จ�ดศู�นี้ย,กัลาง และคว่ามยาว่ของร�ศูม�ของว่งกัลมนี้��
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……………………………………………………………………………………………………….6. จงหาคา y ถ้!ากั.าหนี้ดให! (4 , y)
อย�หางจากัจ�ด (-5 , 2) และ (13 , -6) เป�นี้ระยะเทากั�นี้
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7. R (-7 , -5) , S (3 , 7) , T (6 , 1) เป�นี้จ�ดยอดของร�ปสามเหล��ยมร�ปหนี้*�ง จงหาคว่ามยาว่ของเส!นี้ม�ธยฐานี้จากัจ�ด S ไปย�งด!านี้ RT
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8. กั.าหนี้ดให! A (2 , 7) , B (-5 , 6) เป�นี้จ�ดปลายของสว่นี้ของเส!นี้ต่รง จงหาพิ#กั�ดของจ�ดบนี้สว่นี้ของเส!นี้ต่รงนี้�� ซึ่*�งอย�หางจากั A
เทากั�บ 34 ของระยะทางระหว่าง A และ B
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แบบฝึ�กหั�ดทั�� 3
1. จงหาคว่ามชื่�นี้ของเส!นี้ต่รงท��ผ่านี้จ�ดสองจ�ดต่อไปนี้�� 1.1 (0 , 0) และ (2 , 6)
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1.2 (0 , 0) และ (-2 , 6)
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1.3 (5 , 3) และ (12 , 7)
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1.4 (0 , 0) และ (-3 , -4)
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1.5 (3 , -8) และ (-5 , 7)
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1.6 (t + 1 , s) และ (2t , s - 3)
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……………………………………………………………………………………………………….2. จงหาคา X ท��ท.าให!เส!นี้ต่รงท��ผ่านี้จ�ด P และจ�ด Q ม�คว่ามชื่�นี้เทากั�บ m ต่ามท��กั.าหนี้ดให!
2.1 P (5 , 2) และ Q (x , 6) ; m = 4
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2.2 P (4 , x) และ Q (-3 , 1) ; m = 12
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2.3 P (6 , -3) และ Q (9 , x) ; m = - 23
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2.4 P (x , 12) และ Q (5 , 12) ; m = 0
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2.5 P (1 , x) และ Q (4 , 3) ; m = 43
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……………………………………………………………………………………………………….3. จงหาคว่ามชื่�นี้ของเส!นี้ต่รงท��ผ่านี้จ�ด (a , ab ) และ (b , ba ) เม%�อ a 0 , b 0 และ a b
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……………………………………………………………………………………………………….4. จงหาคว่ามชื่�นี้และคว่ามยาว่ของแต่ละด!านี้ของร�ปสามเหล��ยมซึ่*�งม�จ�ด A (2 ,10) , B (5 , 7) และ C (2 , 4) เป�นี้จ�ดยอด
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……………………………………………………………………………………………………….5. กั.าหนี้ดให! P (-6 , 4) , Q (1 , 4) , R (-1 , -1) และ S ( -
8 , -1) เป�นี้จ�ดยอดของร�ปสามเหล��ยม จงหาคว่ามชื่�นี้ของสว่นี้ของเส!นี้ต่รงแต่ล!ะเส!นี้ซึ่*�งแบงร�ปส��เหล��ยมนี้��ออกัเป�นี้ร�ปสามเหล��ยมสองร�ปท��ม�พิ%�นี้ท��เทากั�นี้
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6. กั.าหนี้ดให! A (-6 , -2) , B (2 , -2) , C และ D เป�นี้จ�ดยอดของร�ปส��เหล��ยมคางหม� ม�ด!านี้ AB เป�นี้ฐานี้ ท��ยาว่เป�นี้ 2 เทาของด!านี้ค�ขนี้านี้ DC ม�ม�ม A เป�นี้ม�มฉากั และม�พิ%�นี้ท�� 24 ต่ารางหนี้ว่ย จงหาคว่ามชื่�นี้ของด!านี้ BC
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แบบฝึ�กหั�ดทั�� 4
1. จงแสดงว่าเส!นี้ต่รงซึ่*�งผ่านี้จ�ด (-2 , -4) และ (3 , 3) ขนี้านี้กั�บเส!นี้ต่รงซึ่*�งผ่านี้จ�ด (1 , -2) และ (6 ,5)
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……………………………………………………………………………………………………….2. ถ้!าเส!นี้ต่รงผ่านี้จ�ด (k , 7) และ (-3 , -2) ขนี้านี้กั�บเส!นี้ต่รงซึ่*�งผ่านี้จ�ด (3 , 2) และ (1 , -4)
จงหาคาของ k
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……………………………………………………………………………………………………….3. จงแสดงว่า (-2 , -1) , (1, 0) , (4 , 3) และ (1 , 2)
เป�นี้จ�ดยอดของร�ปส��เหล��ยมด!านี้ขนี้านี้
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……………………………………………………………………………………………………….4. จ�ด (1 , 2) , (6 , 7) และ (-3 , 4) อย�บนี้เส!นี้ต่รงเด�ยว่กั�นี้หร%อไม เพิราะเหต่�ใด
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5. จงหาคา b ท��ท.าให!จ�ด (b , 6) , (-1 , 4) และ (-4 , 2) อย�บนี้เส!นี้ต่รงเด�ยว่กั�นี้
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……………………………………………………………………………………………………….6. กั.าหนี้ดให! P (-3 , 2) , Q (1 , 6) , R (5 , 4) และ S (3 ,
0) เป�นี้จ�ดยอดของร�ปส��เหล��ยม PQRS จงแสดงว่าจ�ดกั*�งกัลางของด!านี้ท��งส��ของร�ปส��เหล��ยมนี้��เป�นี้จ�ดยอดของร�ปส��เหล��ยมด!านี้ขนี้านี้
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แบบฝึ�กหั�ดทั�� 5
1. เส!นี้ต่รง l ม�คว่ามชื่�นี้ 34 เส!นี้ต่รงท��ต่� �งฉากักั�บเส!นี้ต่รง l จะม�คว่ามชื่�นี้เทาไร
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……………………………………………………………………………………………………….2. จงหาคว่ามชื่�นี้ของเส!นี้ต่รงซึ่*�งต่��งฉากักั�บเส!นี้ต่รงซึ่*�งผ่านี้จ�ด (3 , 4)
และ (-3 , -5)
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……………………………………………………………………………………………………….3. เส!นี้ต่รงซึ่*�งผ่านี้จ�ด (4 , 3) และ (-3 , -5) ต่��งฉากักั�บเส!นี้ต่รงซึ่*�งผ่านี้จ�ด (-2 , -3) และ (-8 , 2) หร%อไม
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……………………………………………………………………………………………………….4. ถ้!าจ�ด P ม�พิ#กั�ดเป�นี้ (a ,b) และจ�ด Q ม�พิ#กั�ดเป�นี้ (-b , a)
โดยท�� a2 + b2 0 จงแสดงว่าสว่นี้ของเส!นี้ต่รง OP ต่��งฉากักั�บสว่นี้ของเส!นี้ต่รง OQ เม%�อ O เป�นี้จ�ดกั.าเนี้#ด
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……………………………………………………………………………………………………….5. จ�ด (1 , 6) , (8 , 8) และ (-7 , 2) เป�นี้จ�ดยอดของร�ปสามเหล��ยมม�มฉากัหร%อไม
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……………………………………………………………………………………………………….6. เส!นี้ต่รงซึ่*�งผ่านี้จ�ด (k , 7) และ (-3 , -2) ต่��งฉากักั�บเส!นี้ต่รงซึ่*�งผ่านี้จ�ด (3 , 2) และ (1 , -4)
จงหาคาของ k
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7. จ�ด (2 , 5) , (2 , 9) , (6 , 9) และ (6 , 5) เป�นี้จ�ดยอดของร�ปเหล��ยม จงแสดงว่าเส!นี้ทแยงม�มท��งสอง ของร�ปส��เหล��ยมต่��งฉากัซึ่*�งกั�นี้และกั�นี้
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……………………………………………………………………………………………………….8. กั.าหนี้ดจ�ด A (6 , 8) , B (5 , 4) , C (3 , 6) และ D (4 ,
10) จงพิ#จารณาว่า 8.1 ABCD เป�นี้ร�ปส��เหล��ยมด!านี้ขนี้านี้หร%อไม8.2 ABCD เป�นี้ร�ปส��เหล��ยมม�มฉากัหร%อไม
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แบบฝึ�กหั�ดทั�� 6
1. ถ้!า t = {(x , y) x – 2y = 4 } จงพิ#จารณาว่าจ�ดต่อไปนี้��เป�นี้สมาชื่#กัของคว่ามส�มพิ�นี้ธ, t หร%อไม
1.1 (1 , 0)
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1.2 (-8 , 6)
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1.3 (0 , -2)
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1.4 (2 , -1)
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1.5 (6√2 , √2)
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2. จงหาคว่ามส�มพิ�นี้ธ,ซึ่*�งม�กัราฟเป�นี้เส!นี้ต่รงต่ามสมบ�ต่#ท��กั.าหนี้ดให!ต่อไปนี้��
2.1 ขนี้านี้กั�บแกันี้ X และอย�เหนี้%อแกันี้ X เป�นี้ระยะ 37 หนี้ว่ย
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2.2 ขนี้านี้กั�บแกันี้ Y และอย�ทางซึ่!ายแกันี้ Y เป�นี้ระยะ 23 หนี้ว่ย
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2.3 ขนี้านี้กั�บแกันี้ X และอย�หางจากัจ�ด (0 , 3) เป�นี้ระยะ 4 หนี้ว่ย
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2.4 ขนี้านี้กั�บแกันี้ Y และอย�หางจากัจ�ด (-2 , 0) เป�นี้ระยะ 5 หนี้ว่ย
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……………………………………………………………………………………………………….3. จงบอกัคว่ามชื่�นี้และระยะต่�ดแกันี้ X และแกันี้ Y ของกัราฟของแต่ละสมกัาร
สมการ์ ควิามช�น ร์ะยะต�ดแกน X ร์ะยะต�ดแกน Y1) 2x – 3y = 72) 5x + 4y – 2 = 03) x – 4y + 5 = 04) 3x + 2y + 7 = 05) 5x – y – 11 = 0
6) 34x - 43 y
= 247) x – y = 08) 2y + 3 = 09) x = 4 10) 3 (y -1) = -2 (x – 2)
4. จงแสดงว่าเส!นี้ต่รง 3y = 2x – 6 ขนี้านี้กั�บเส!นี้ต่รง y = 23
x + 1
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……………………………………………………………………………………………………….5. จงแสดงว่าเส!นี้ต่รง 2x + y = 8 ต่��งฉากักั�บเส!นี้ต่รง y = 12
x - 5
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……………………………………………………………………………………………………….6. จงหาสมกัารของเส!นี้ต่รงท��ผ่านี้จ�ด (7 , 5) และขนี้านี้กั�บเส!นี้ต่รง x + 2y + 12 = 0
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……………………………………………………………………………………………………….7. จงหาสมกัารของเส!นี้ต่รงท��ผ่านี้จ�ด (3 , 2) และต่��งฉากักั�บเส!นี้ต่รง 3x - 2y + 12 = 0
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แบบฝึ�กหั�ดทั�� 7
1. จงหาระยะหางระหว่างกัราฟของสมกัารกั�บจ�ดท��กั.าหนี้ดให!ต่อไปนี้��1.1 6x - 8y + 4 = 0 , (2 , -3)
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1.2 4x + 3y - 8 = 0 , (0 , 6)
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1.3 2x + 3y = 13 , (0 , 0)
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1.4 y - 4 = 75 (x – 3) , (8 , 11)
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1.3 y = 1 , (-1 , 1)
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……………………………………………………………………………………………………….2. จงหาระยะหางระหว่างกัราฟของสมกัารแต่ละค�ต่อไปนี้��
2.1 3x + 4y - 7 = 0 , 3x + 4y + 3 = 0
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2.2 3x - 4y - 7 = 0 , 6x - 8y + 16 = 0
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2.3 5x + 12y - 15 = 0 , 10x + 24y + 9 = 0
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2.4 x - y - 3 = 0 , 3x - 3y + 7 = 0
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2.5 3x + y + 5 = 0 , 7 - 3x - y = 0
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……………………………………………………………………………………………………….3. จงหาสมกัารของเส!นี้ต่รงท��ขนี้านี้กั�บเส!นี้ต่รง 3x - 4y - 5 =
0 และอย�หางจากัเส!นี้ต่รงนี้�� 1 หนี้ว่ย
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……………………………………………………………………………………………………….4. จงหาสมกัารของเส!นี้ต่รงท��ขนี้านี้กั�บเส!นี้ต่รง 4x - 3y + 26
= 0 และอย�หางจากัจ�ด (8 , 8) 2 หนี้ว่ย
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……………………………………………………………………………………………………….5. จงหาสมกัารของเส!นี้ต่รงท��ต่� �งฉากักั�บเส!นี้ต่รง 12y = 5x - 7
และอย�หางจากัจ�ด (-1 , 2) 3 หนี้ว่ย
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6. ถ้!าเส!นี้ต่รง 12x - 5y - 10 = 0 เป�นี้เส!นี้ต่รงท��อย�กั*�งกัลางระหว่างเส!นี้ขนี้านี้ค�หนี้*�ง ซึ่*�งอย�หางกั�นี้ 8 หนี้ว่ย จงหาสมกัารของเส!นี้ขนี้านี้ค�นี้��
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แบบฝึ�กหั�ดทั�� 8
1. จงหาสมกัารของว่งกัลมท��สอดคล!องกั�บเง%�อนี้ไขต่อไปนี้�� 1.1 จ�ดศู�นี้ย,กัลางอย�ท�� (2 , 1) และร�ศูม�ยาว่ 3 หนี้ว่ย
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1.2 จ�ดศู�นี้ย,กัลางอย�ท��จ�ดกั.าเนี้#นี้ และผ่านี้จ�ด (7 , 4)
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1.3 จ�ดศู�นี้ย,กัลางอย�ท�� (-1 , 5) และ ผ่านี้จ�ด (-4 ,-6)
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1.4 จ�ดปลายของเส!นี้ผ่านี้ศู�นี้ย,กัลางสค%อ (-1 , 3) และ (7 , -5)
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1.5 จ�ดศู�นี้ย,กัลางอย�ท�� (7 , -3) และส�มผ่�สกั�บแกันี้ X
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1.6 ว่งกัลมอย�ในี้คว่อดร�นี้ต่,ท�� 1 และส�มผ่�สแกันี้ X และแกันี้ Y
ร�ศูม�ยาว่ 5 หนี้ว่ย
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2. จงแสดงว่าสมกัารต่อไปนี้��เป�นี้สมกัารของว่งกัลม แล!ว่หาจ�ดศู�นี้ย,กัลางและคว่ามยาว่ของร�ศูม�ของว่งกัลม
2.1 x2 + y2 + 4x - 2y + 1 = 0
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2.2 x2 + y2 + y = 0
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2.3 x2 + y2 + 10x - 4y + 13 = 0
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2.4 x2 + y2 + x + 2y + 1 = 0
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2.5 x2 + y2 + 6x + 2 = 0
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2.6 x2 + y2 - 12 x + 12y - 18 = 0
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3. จงหาสมกัารว่งกัลม ซึ่*�งม�จ�ดศู�นี้ย,กัลางอย�ท��จ�ด (-1 , 1) และส�มผ่�สกั�บเส!นี้ต่รงซึ่*�งม�สมกัารเป�นี้ 3x – 2y + 18 = 0
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……………………………………………………………………………………………………….4. จงหาจ�ดต่�ดของกัราฟของสมกัาร x + y = 1 และ x2 + y2 - x -3y = 0
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……………………………………………………………………………………………………….5. จงหาสมกัารเส!นี้ส�มผ่�สของว่งกัลมท��ม�สมกัารเป�นี้ x2 + y2 -
10x = 0 ณ จ�ดท��ว่งกัลมต่�ดเส!นี้ต่รง 4x + 3y = 20
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……………………………………………………………………………………………………….6. จงเข�ยนี้กัราฟของอาณาบร#เว่ณซึ่*�งกั.าหนี้ดโดยเซึ่ต่ต่อไปนี้��
6.1 {(x , y) x2 + y2 4 }
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6.2 {(x , y) x2 + y2 > 9 }
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……………………………………………………………………………………………………….7. จงหาพิ%�นี้ท��ของบร#เว่ณท��อย�นี้อกัว่งกัลม x2 + y2 = 4 แต่อย�ภายในี้ว่งกัลม x2 + y2 - 4y - 12 = 0
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……………………………………………………………………………………………………….8. จงหาสมกัารของว่งกัลมท��ม�จ�ดศู�นี้ย,กัลาง (2 , -5) และร�ศูม�ยาว่ 2 หนี้ว่ย
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……………………………………………………………………………………………………….9. จงหาสมกัารว่งกัลมท��ม�จ�ดศู�นี้ย,กัลาง (-5 , 1) และผ่านี้จ�ดกั.าเนี้#ด
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……………………………………………………………………………………………………….10. จงหาสมกัารของว่งกัลมท��ม� P(2 , 3) และ Q (-1 , 8) เป�นี้จ�ดปลายของเส!นี้ผ่านี้ศู�นี้ย,กัลาง
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แบบฝึ�กหั�ดทั�� 9
1. จงหาจ�ดยอด โฟกั�ส คว่ามเย%�องศู�นี้ย,กัลาง คว่ามยาว่ของแกันี้เอกัและแกันี้โท ของว่งร�แล!ว่เข�ยนี้กัราฟ
1.1 x2
9+ y
2
25 = 1
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1.2 x2
25+ y
2
9 = 1
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1.3 9x2 + 4y2 = 36
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1.4 4x2 + 25y2 = 100
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1.5 x2 + 4y2 = 16
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1.6 4x2 + y2 = 16
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1.7 2x2 + y2 = 3
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1.8 5x2 + 6y2 = 30
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1.9 x2 + 4y2 = 1
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1.10 9x2 + 4y2 = 1
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1.11 12 x2 + 18 y
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1.12 x2 = 4 - 2y2
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1.13 y2 = 1 - 2x2
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1.14 20x2 + 4y2 = 5
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2. จงหาสมกัารของว่งร�ท��สอดคล!องกั�บเง%�อนี้ไขท��กั.าหนี้ดให!2.1 โฟกั�ส (-4 , 0) , (4 , 0) จ�ดยอด (-5 , 0) , (5 ,
0)
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2.2 โฟกั�ส (0 , -3) , (0 , 3) จ�ดยอด (0 , -5) , (0 , 5)
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2.3 แกันี้เอกัยาว่ 4 หนี้ว่ย แกันี้โทยาว่ 2 หนี้ว่ย โฟกั�สอย�บนี้แกันี้ Y
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2.4 แกันี้เอกัยาว่ 6 หนี้ว่ย แกันี้โทยาว่ 4 หนี้ว่ย โฟกั�สอย�บนี้แกันี้ X
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2.5 โฟกั�ส (0 , -2) , (0 , 2) แกันี้โทยาว่ 6 หนี้ว่ย
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2.6 โฟกั�ส (-5 , 0) , (5 , 0) แกันี้เอกัยาว่ 12 หนี้ว่ย
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2.7 จ�ดปลายแกันี้เอกั (-10 , 0) , (10 , 0) ระยะหางระหว่างโฟกั�สเทากั�บ 6 หนี้ว่ย
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2.8 จ�ดปลายแกันี้โท (0 , -3) , (0 , 3) ระยะหางระหว่างโฟกั�สเทากั�บ 8 หนี้ว่ย
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2.9 แกันี้เอกัยาว่ 10 หนี้ว่ย โฟกั�สอย�บนี้แกันี้ X ว่งร�ผ่านี้จ�ด (√5 , 2)
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2.10 คว่ามเย%�องศู�นี้ย,กัลาง 19 โฟกั�ส (0 , -2) , (0 , 2)
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2.11 คว่ามเย%�องศู�นี้ย,กัลาง 0.8 โฟกั�ส (-1.5 , 0) , (1.5 , 0)
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2.12 คว่ามเย%�องศู�นี้ย,กัลาง √32
โฟกั�สอย�บนี้แกันี้ Y แกันี้เอกัยาว่ 4 หนี้ว่ย
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……………………………………………………………………………………………………….3. จงหาจ�ดศู�นี้ย,กัลาง โฟกั�ส และจ�ดยอดของว่งร� และหาคว่ามยาว่ของแกันี้เอกัและแกันี้โทแล!ว่เข�ยนี้กัราฟ
3.1 (x−1)2
4 + ( y−2)
2
9 = 1
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3.2 (y + 3)2 + (x−3)2
16 = 1
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3.3 y2
9 + (x+5)
2
25 = 1
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3.4 x2 + ( y+2)2
4 = 1
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……………………………………………………………………………………………………….4. ดาว่เคราะห,โคจรรอบดว่งอาท#ต่ย,โดยท��ดว่งอาท#ต่ย,อย�ท��โฟกั�สจ�ดหนี้*�ง จ�ดท��ดาว่เคราะห,อย�ใกัล!ดว่งอาท#ต่ย,มากัท��ส�ด เร�ยกัว่า perihelion
และจ�ดท��ดาว่เคราะห,อย�ไกัลดว่งอาท#ต่ย,มากัท��ส�ด เร�ยกัว่า aphelion
จ�ดท��งสองนี้��เป�นี้จ�ดยอดของว่งโคจร โลกัอย�หางจากัดว่งอาท#ต่ย, 147,000,000 กั#โลเมต่รท�� perihelion และ 153,000,000
กั#โลเมต่ร ท�� aphelion จงหาสมกัารของว่งโคจรของโลกั รอบดว่งอาท#ต่ย, (ให!จ�ดกั.าเนี้#ดอย�ท��จ�ดศู�นี้ย,กัลางของว่งโคจร)
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……………………………………………………………………………………………………….5. ว่งโคจรของดาว่พิล�โต่รอบดว่งอาท#ต่ย,เป�นี้ว่งร�ท��ม�คว่ามเย%�องศู�นี้ย,กัลางมากัท��ส�ด ค%อ 0.25 คว่ามยาว่ของแกันี้โทของว่งโคจรเทากั�บ 10,000,000,000 กั#โลเมต่ร จงหาระยะทางระหว่าง ดว่งพิล�โต่กั�บดว่งอาท#ต่ย,ท�� perihelion และท�� aphelion
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แบบฝึ�กหั�ดทั�� 11
1. จงหาโฟกั�ส ไดเรกัต่ร#กัซึ่, และคว่ามยาว่ของเลต่�สเรกัต่�มของพิาราโบลา แล!ว่เข�ยนี้กัราฟ
1.1 y = 4x2
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1.2 x = y2
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1.3 x2 = 9y
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1.4 y = 3x2
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1.5 y2 = 5x
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1.6 y2 = -2x
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1.7 x2 = -8y
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1.8 x = 12 y2
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1.9 x + 6y2 = 0
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1.10 x2 - 7y = 0
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1.11 5x2 + 3y = 0
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1.12 8x + 12y2 = 0
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2. จงหาสมกัารของพิาราโบลาท��ม�จ+ดยอดอย*1ทั��จ+ดก,าเน�ดและสอดคล!องกั�บเง%�อนี้ไขต่อไปนี้��
2.1 โฟกั�ส ค%อ F (2 , 0)
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2.2 โฟกั�ส ค%อ F (- 12 , 0)
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2.3 โฟกั�ส ค%อ F (0, -8)
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2.4 โฟกั�ส ค%อ F (0, 5)
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2.5 ไดเรกัต่ร#กัซึ่, ค%อ y = 2
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2.6 ไดเรกัต่ร#กัซึ่, ค%อ x = 6
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2.7 ไดเรกัต่ร#กัซึ่, ค%อ x = -10
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2.8 ไดเรกัต่ร#กัซึ่, ค%อ y = - 12
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2.9 โฟกั�สอย�บนี้แกันี้ X ด!านี้บว่กั หางจากัไดเรกัต่ร#กัซึ่, 2
หนี้ว่ย
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2.10 ไดเรกัต่ร#กัซึ่,ม�ระยะต่�ดแกันี้ Y เทากั�บ 6 หนี้ว่ย
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2.11 เป�นี้โค!งเป7ดด!านี้บนี้ ม�โฟกั�สหางจากัจ�ดยอด 8 หนี้ว่ย
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2.12 คว่ามยาว่ของเลต่�สเรกัต่�มเทากั�บ 8 หนี้ว่ย และโฟกั�สอย�บนี้แกันี้ Y ด!านี้ลบ
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3. จงหา จ�ดยอด โฟกั�ส และไดเรกัต่ร#กัซึ่,ของพิาราโบลา แล!ว่เข�ยนี้กัราฟ
3.1 (x – 3)2 = 8(y + 2)
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3.2 (y – 5)2 = -6x + 12
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3.3 -4(x + 12)2 = y
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3.4 y2 = 16x + 8
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……………………………………………………………………………………………………….4. ว่#ศูว่กัรออกัแบบจานี้ร�บส�ญญาณดาว่เท�ยม โดยให!พิ%�นี้ท��ผ่#ว่ท��ร �บส�ญญาณเป�นี้ทรงพิาราโบลา (ร�ปเรขาคณ#ต่สามม#ต่#ท��เกั#ดจากัพิาราโบลาหม�นี้รอบแกันี้สมมาต่ร ม�ภาคต่�ดขว่างท��ผ่านี้แกันี้สมมาต่รเป�นี้พิาราโบลา) จานี้กัว่!าง 15 ฟ�ต่ และล*กั 23 ฟ�ด (ด�ร�ป)
4.1 ถ้!ากั.าหนี้ดระบบพิ#กั�ดฉากัให!จ�ดกั.าเนี้#ดอย�ท��จ�ดยอดและแกันี้ X
ซึ่!อนี้กั�บแกันี้ของพิาราโบลา แล!ว่จงหาสมกัารของพิาราโบลา4.2 คว่รต่#ต่ต่��งอ�ปกัรณ,รว่มส�ญญาณไว่!ในี้ต่.าแหนี้งใด หางจากัจ�ด
ยอดเทาใด
อ�ปกัรณ,รว่มส�ญญาณ
23 ฟ�ต่
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แบบฝึ�กหั�ดทั�� 12
1. จงหาจ�ดยอด โฟกั�ส และเส!นี้กั.ากั�บของไฮเพิอร,โบลาแล!ว่เข�ยนี้กัราฟ1.1 x2 - y
2
9 = 1
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1.2 y2
4 - x
2
16 = 1
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1.3 y2 - x2 = 1
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1.4 x2 - 4y2 - 8 = 0
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1.5 y2
4 - x2 = 1
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1.6 x2
9 - y
2
16 = 1
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1.7 9y2 - 4x2 = 36
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1.8 y2 - 2x2 = 3
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1.9 16x2 - y2 = 144
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1.10 x2 - y2
25 = 1
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1.11 25y2 - 9x2 = 225
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1.12 y2 - 4x2 + 1 = 0
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1.13 9y2 - 25x2 = 225
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1.14 y2
2 - x2 = 1
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1.15 y2 - x2 - 4 = 0
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1.16 9y2 - 16x2 = 1
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2. จงหาสมกัารของไฮเพิอร,โบลาท��สอดคล!องกั�บเง%�อนี้ไขต่อไปนี้�� 2.1 โฟกั�ส ค%อจ�ด (-5 , 0) , (5 , 0) และจ�ด (0 , -3) , (0 ,
3) เป�นี้จ�ดยอด
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2.2 โฟกั�ส ค%อจ�ด (-10 , 0) , (10 , 0) และจ�ด (-8 , 0) ,
(8 , 0) เป�นี้จ�ดยอด
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2.3 โฟกั�ส ค%อจ�ด (-2 , 0) , (2 , 0) และจ�ด (-1 , 0) , (1 ,
0) เป�นี้จ�ดยอด
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2.4 โฟกั�ส ค%อจ�ด (0 , -6) , (0 , 6) และจ�ด (0 , -2) , (0 ,
2) เป�นี้จ�ดยอด
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2.5 จ�ดยอด ค%อจ�ด (-1 , 0) , (1 , 0) เส!นี้กั.ากั�บ ค%อ y = 5x
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2.6 จ�ดยอด ค%อจ�ด (0 , -6) , (0 , 6) เส!นี้กั.ากั�บ ค%อ y =
13 x
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2.7 โฟกั�ส ค%อจ�ด (0 , -8) , (0 , 8) เส!นี้กั.ากั�บ ค%อ y =
12 x
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2.8 จ�ดยอด ค%อจ�ด (0 , -6) , (0 , 6) ไฮเพิอร,โบลาผ่านี้จ�ด (-5 , 9)
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2.9 เส!นี้กั.ากั�บ ค%อ y = x ไฮเพิอร,โบลาผ่านี้จ�ด (5 , 3)
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2.10 โฟกั�ส ค%อจ�ด (-3 , 0) , (3 , 0) ไฮเพิอร,โบลาผ่านี้จ�ด (4 , 1)
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2.11 โฟกั�ส ค%อจ�ด (-5 , 0) , (5 , 0) แกันี้ต่ามขว่างยาว่ 6 หนี้ว่ย
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2.12 โฟกั�ส ค%อจ�ด (0 , -1) , (0 , 1) แกันี้ต่ามขว่างยาว่ 1 หนี้ว่ย
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3. จงหาจ�ดศู�นี้ย,กัลาง โฟกั�ส จ�ดยอด และเส!นี้กั.ากั�บของไฮเพิอร,โบลา แล!ว่เข�ยนี้กัราฟ
3.1 (x−1)2
9 - ( y+3)
2
16 = 1
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3.2 (x + 8)2 - (y – 6)2 = 1
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3.3 y2 - (x−1)2
4 = 1
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3.4 ( y−1)2
25 - (x – 3)2 = 1
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แบบฝึ�กหั�ดทั�� 13
จงหาว่าสมกัารต่อไปนี้��ม�กัราฟเป�นี้ภาคต่�ดกัรว่ยชื่นี้#ดใด จงหาโฟกั�สและจ�ดยอด (ถ้!าม�) แล!ว่เข�ยนี้กัราฟ
1) y2
12 + x = 1
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2) y2
12 + x
2
144 = y12
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3) x2 - y2 + 144 = 0
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4) x2 + 6x = 9y2
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5) 4x2 + y2 = 8(x + y)
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6) 3x2 - 6(x + y) = 10
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7) x = y2 - 16y
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8) 2x2 + 4 = 4x + y2
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9) 2x2 - 12x + y2 + 6y + 26 = 0
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10) 36x2 - 4y2 - 36x - 8y = 31
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11) 9x2 + 8y2 - 15x + 8y = -27
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12) x2 + 4y2 = 4x + 8
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