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接線の解法 | 夢を叶える塾
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接線の問題の解法

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曲線の接線を求める問題です。

基本問題

(1)y=3x^2-2x+4x=1における接線の方程式を求めなさい
(2)(-1,-3)からy=3x^2-2x+4に引いた接線の方程式を求めなさい

解き方

y=f(x)としたときの微分係数f'(a)x=aの接線の傾きでした。
一次関数なので、後もう一つ何か情報があれば解けそうですね。

接線の公式がありますので、できればこちらも覚えてしまいましょう。
闇雲に覚えるのは難しいので、意味を理解しながら覚えてくださいね。
y-f(a)=f'(a)(x-a)

解説

(1)y=3x^2-2x+4x=1における接線の方程式を求めなさい
説明のためy=f(x)=3x^2-2x+4とします。
まずは微分しましょう。
f'(x)=6x-2
x=1における接線の傾きはf'(1)=6-2=4です。
接線は直線なので一次関数で表せます。
直線の方程式はy=4x+bの形です。
(1,f(1))を接線は通るので、これを直線の方程式に代入して
3-2+4=4+b
b=1
という事で接線はy=4x+1

(2)(-1,-3)からy=3x^2-2x+4に引いた接線の方程式を求めなさい
(1)は曲線上の点での接線でしたが、(2)は曲線上に無い点からの接線で、少し難しくなりますね
説明のためy=f(x)=3x^2-2x+4とします。

問題を解こうにも果たしてどこで接するのかわかりません。
というわけで接する点を(p,f(p))としましょう。
この点の接線が(-1,-3)を通ってくれればいいですね。

(p,f(p))の接線の方程式y=ax+bを考えましょう。
微分係数f'(p)=6p-2が傾きになりますのでy=(6p-2)x+bとなります。
(p,f(p))をとおるので、
f(p)=(6p-2)p+b
3p^2-2p+4=(6p-2)p+b
b=3p^2-2p+4-(6p-2)p
b=-3p^2+4
という事で、y=(6p-2)x-3p^2+4となります。

これが(-1,-3)を通るという事で代入してみましょう。
-3=(6p-2)(-1)-3p^2+4
このpの二次方程式を解けば良さそうですね。

解いていきます。
-3=(6p-2)(-1)-3p^2+4
0=-6p-3p^2+9
p^2+2p-3=0
(p+3)(p-1)=0
という事でp=-3,1が得られました。
接する点が(p,f(p))でしたから、(-3,37),(1,5)で接します。

(1,5)で接するときの接線の方程式は、(1)でやりましたね。
(-3,37)で接するときの接線の方程式も同様にして求めましょう。

応用問題

f(x)=x^2-2x+5,g(x)=-x^2+4x-4のとき、f(x),g(x)それぞれに接する接線の方程式を求めなさい

解き方

共通する接線を求める問題です。
基本問題の組み合わせです。
まずは問題に書いてある事を整理して答えを導く数学の解き方の王道で解いてみましょう。

解説

f(x)=x^2-2x+5,g(x)=-x^2+4x-4のとき、f(x),g(x)それぞれに接する接線の方程式を求めなさい
まず問題に書いてあることから考えていこうと思うと、それぞれの接線の方程式が知りたいですね。
という事で、P(p,f(p))Q(q,g(q))のそれぞれの接線の方程式l(x),m(x)を求めてみましょう。

f'(x)=(x^2-2x+5)'
f'(x)=2x-2
l(x)=(2p-2)x+b
p^2-2p+5=(2p-2)p+b
p^2-2p+5=2p^2-2p+b
p^2-2p^2-2p+2p+5=b
b=-p^2+5
l(x)=(2p-2)x-p^2+5

g'(x)=(-x^2+4x-4)'
g'(x)=-2x+4
m(x)=(-2q+4)x+c
-q^2+4q-4=(-2q+4)q+c
-q^2+4q-4=-2q^2+4q+c
-q^2+2q^2+4q-4q-4=c
c=q^2-4
m(x)=(-2q+4)x+q^2-4

さて、共通接線だそうです。
という事は、傾きと切片が等しくなるという事ですね。
2p-2=-2q+4・・・①
-p^2+5=q^2-4・・・②
2変数に2つの式なので、この連立方程式を解いてみましょう。

①からp=-q+3を②に代入して、
-(-q+3)^2+5=q^2-4
-q^2+6q-9+5=q^2-4
-q^2-q^2+6q-4+4=0
-2q^2+6q=0
q^2-3q=0
q(q-3)=0
q=0,3
①に代入して
p=3,0

という事で、共通の接線と接点を答えていきます。
p=3,q=0のとき
l(x)=(2p-2)x-p^2+5m(x)=(-2q+4)x+q^2-4p,qをそれぞれ代入して、
l(x)=m(x)=4x-4
f(x),g(x)x=p,qをそれぞれ代入して接点を求めると、
f(x)上の接点(3,8)g(x)上の接点(0,-4)

p=0,q=3のとき
同様に、求める事が出来て、
l(x)=m(x)=-2x+5
f(x)上の接点(0,5)g(x)上の接点(3,-1)

さて、もう一つ解法をご紹介します。
1つの接線を求めた後、接線と曲線が接するという条件を確認する方法です。
判別式を用いる方法ですね。

f(x)の接線のl(x)=y=(2p-2)x-p^2+5は先に求めたものを拝借します。
これがg(x)=y=-x^2+4x-4に接するという条件を確認してみましょう。
どうやって確認するか?
2次が接するという事は、交点が重解という事でしたね。
交点を求めるためには、連立方程式です。

代入法でyを消去します。
-x^2+4x-4=(2p-2)x-p^2+5
-x^2+4x-(2p-2)x-4+p^2-5=0
-x^2+(6-2p)x+p^2-9=0
これが重解を持つという事は判別式が0になるという事なので、
D=(6-2p)^2+4(p^2-9)=0
36-24p+4p^2+4p^2-36=0
8p^2-24p=0
p^2-3p=0
p(p-3)=0
ということでp=3,0
f(x)上の接点がわかり、接線も求める事ができます。

この解法の場合、ここまでは割と早く求める事ができます。
ので、共通接線を求める場合は、この解法の方が少しだけ早いかもしれませんね。
思いつかなくても、最初の解き方を貫いた方がかえって早いかもしれませんが。

なお、今回の問題は接点まで求める必要があります。
l(x),g(x)の交点を求める式は既に導いています。
求めたpを代入した二次方程式を解いて、g(x)上の交点のx座標を求めましょう。

終わりに

微分係数が接線の傾きであるという性質さえ覚えておけば、あとは一次関数の問題になります。
公式も確かに早く計算できるかもしれませんが、公式に頼るよりも公式を使わずに解ける力の方が重要だと私は思います。

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