Re: [微積] 兩曲線公切線

[mack (腦海裡依然記得妳)]

※ 引述《stranger522 (香菇勇)》之銘言：
: y=x^2
: y=-x^2+10x-17
: 求兩曲線的公切線
: 不知怎麼下手..有請各位高手~
y = x^2
y = -(x-5)^2+8
公切線:(y-4) = m(x-5/2) => y = mx - 5m/2 + 4 與 y = x^2 交於一點
=> x^2 = mx - 5m/2 + 4 => x^2 - mx + (5m/2 - 4) = 0 唯一解
=> D = (-m)^2 - 4 * 1 * (5m/2 - 4) = 0
=> D = m^2 - 10m + 16 = (m - 2)(m - 8) = 0
=> m = 2 or 8
=> 公切線:y = 2x-1 or y = 8x-16

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推 stranger522 :請問 公切線:(y-4) = m(x-5/2) 是怎麼設的呢?
→ mack        :公切線必過兩頂點的中點
推 stranger522 :恕我冒昧..公切線必過兩頂點中點是什麼理論呢?
→ mack        :因為y=x^2對(a,b)的點對稱圖形y=-(x-5)^2+8
→ mack        :(a,b)為對稱兩點的中點  我取兩頂點