โจทย์ปัญหาเรื่องภาคตัดกรวย ข้อที่ 1

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โจทย์ปัญหาเรื่องภาคตัดกรวย ข้อที่ 1

Sunday, 28 December 2008 23:17

ระดับความยาก 7/10
Ent'42 ให้ S เป็นเซตของจำนวนจริง m ทั้งหมดที่ทำให้เส้นตรง y = mx ตัดกับวงกลม
ขอบเขตบนค่าน้อยสุดของ S คือจำนวนในข้อใดต่อไปนี้

1. 1/2 2. 2/3 3. 3/4 4. 4/5

เฉลย >> ข้อ 3
ข้อนี้ต้องวิเคราะห์สมการวงกลมที่ให้มาก่อนครับ โดยการจัดรูปสมการจะได้ว่า
x^2 - 10x + 25 - 9 + y^2 = 0 (ขอใช้สัญลักษณ์ ^ แทนเครื่องหมายยกกำลังนะครับเพื่อความรวดเร็ว)
(x-5)^2 + y^2 = 3^2 >> ในการจัดรูปสมการวงกลมนี้ใครยังไม่ทราบว่าทำอย่างไีรขอให้ทบทวนเรื่องนี้ใหม่ครับ
ดังนั้นจากสมการวงกลมที่เราจัดรูปแล้วจะได้ว่า เป็นวงกลมที่มีจุดศูนย์กลางอยู่ที่จุด (5,0) และมีรัศมียาว 3 หน่วย
ต่อมาให้วิเคราะห์เส้นตรงที่ตัดสมการวงกลมนี้ตามรูป

ความชันของเส้นบนและเส้นล่างจะมีค่าตัวเลขเท่ากัน ต่างกันเพียงเครื่องหมายอันนึงจะเป็นบวกและอีกอัน
จะเป็นลบเท่านั้น

ขั้นต่อไป เราต้องทำการหาค่า m ซึ่งสามารถทำได้หลายวิธีครับ
(ความจริงมีสูตรหาครับแต่พี่ลืมการใช้สูตรไปแล้วและขอแนะนำว่าเราไม่ควรจำสูตรทุกสูตร
บางสูตรที่เป็นสูตรลัดจะไม่ควรจำ สาเหตุเพราะว่าเมื่อเราเจอโจทย์ที่ประยุกต์มากขึ้นจะไม่สามารถใช้สูตรที่จำมาได้
ดังนั้นเราควรจำขั้นตอนการทำเสียมากกว่า เมื่อเี่ราเจอโจทย์ยากแค่ไหนเีราก็จะสามารถทำได้ครับ)
วิธีการก็คือหาสมการเส้นตรงสีเขียวครับว่าเป็นอย่างไร ซึ่งเราทราบว่าความชันของเส้นตรงสองเส้นที่ตัดกัน
คูณกันจะต้องมีค่าเท่ากับ -1 ดังนัน ความชันของเส้นตรงสีเขียวจะมีค่าเท่ากับ -1/m
ทำให้เราสร้างสมการได้ว่า y = (-1/m)×(x - 5)
ดังนั้นตอนนี้เรามี 3 สมการแล้วก็คือ
1) y = mx
2) y = (-1/m)×(x - 5)
3) (x-5)^2 + y^2 = 3^2
ก็แก้สมการ 3 ตัวแปรออกมาครับ
นำค่า y = mx จากสมการที่ 1) ไปแทนในสมการที่ 2) ได้ว่า
4) mx = (-1/m)×(x - 5)
5) m^2 = (5 - x) / x
นำค่า y = mx จากสมการที่ 1) ไปแทนในสมการที่ 3) ได้ว่า
6) (x-5)^2 + (m^2)(x^2) = 9
นำค่า m^2 = (5 - x) / x จากสมการที่ 5 ไปแทนในสมการที่ 6 ได้ว่า
7) (x-5)^2 + (x^2)(5 - x) / x = 9
(x-5)^2 + (x)(5 - x) = 9
x^2 - 10x + 25 + 5x - x^2 = 9
-5x = -16
x = 16 / 5

นำค่า x ไปแทนในสมการ 5) จะได้ค่า m = 0.75 , -0.75
ซึ่งได้ขอบเขตของช่วง m คือ [-0.75 , 0.75]
เพราะฉะนั้นขอบเขตบนค่าน้อยสุดของ S คือ 0.75 = 3/4

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