Математика для менеджеров. Часть I. Учебное пособие
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ȼɜɟɞɟɦ |
ɜɟɤɬɨɪ |
n (A,B) |
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɣ |
ɢɫɤɨɦɨɣ |
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ɩɪɹɦɨɣ Ɍɨɝɞɚ |
ɢɡ ɭɫɥɨɜɢɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɫɬɢ ɜɟɤɬɨɪɨɜ n ɢ |
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M 0 N ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ |
n M 0 N 0 ȼ ɪɟɡɭɥɶɬɚɬɟ |
ɩɨɥɭɱɚɟɦ |
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ɭɪɚɜɧɟɧɢɟ Ⱥ(ɯ-ɯ0)+ȼ(ɭ-ɭ0 ɢɥɢ |
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(2) |
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Ⱥɯ ȼɭ ɋ=0, |
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ɝɞɟ ɋ=–Ⱥɯ0–ȼɭ0. |
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ɍɪɚɜɧɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ |
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n (A,B) |
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ɨɛɳɢɦ ɭɪɚɜɧɟɧɢɟɦ ɩɪɹɦɨɣ |
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ɧɚ ɩɥɨɫɤɨɫɬɢ |
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ȼɟɤɬɨɪ n (A, B) |
ɧɚɡɵɜɚ |
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Ax By C 0 |
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S |
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ɟɬɫɹ ɧɨɪɦɚɥɶɸ. |
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ɉɪɢɦɟɪ ɇɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱ ɤɭ Q(2, – ɩɚɪɚɥɥɟɥɶɧɨ ɨɫɢ Ɉɭ.
Ɋɟɲɟɧɢɟ ȼ ɤɚɱɟɫɬɜɟ ɧɚɩɪɚɜɥɹɸɳɟɝɨ ɜɟɤɬɨɪɚ S ɦɨɠɧɨ ɜɡɹɬɶ ɜɟɤɬɨɪ j (0,1) ɉɨɞɫɬɚɜɢɜ ɞɚɧɧɵɟ ɜ ɭɪɚɜɧɟɧɢɢ ɩɨɥɭɱɢɦ
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x 2 |
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y 3 |
ɗɬɨ ɤɚɧɨɧɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɨɛɵɱɧɨ ɩɟɪɟɩɢɫɵɜɚɸɬ |
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ɜ ɨɛɳɟɦ ɜɢɞɟ ɯ – 2 = 0 ɢɥɢ ɯ = 2. |
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ɉɪɢ B 0 |
ɨɛɳɟɟ ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ |
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ɜɢɞɟ |
kx + b, ɝɞɟ k A B, |
b C B . |
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ɭ |
(3) |
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ɍɪɚɜɧɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɫ ɭɝɥɨɜɵɦ ɤɨɷɮɮɢ ɰɢɟɧɬɨɦ ɭɝɥɨɜɨɣ ɤɨɷɮɮɢɰɢɟɧɬ k = tg Į ɝɞɟ Į – ɭɝɨɥ ɧɚɤɥɨɧɚ ɩɪɹɦɨɣ ɤ ɨɫɢ Ɉɯ. ɉɪɢ k = 0 (Į = 0)ɭɪɚɜɧɟɧɢɟ ɞɚɟɬ ɩɪɹɦɭɸ ɩɚɪɚɥɥɟɥɶɧɭɸ ɨɫɢ Ɉɯ. ɂɡ ɭɪɚɜɧɟɧɢɹ ɧɟɥɶɡɹ ɩɨɥɭɱɢɬɶ ɭɪɚɜɧɟ ɧɢɟ ɩɪɹɦɨɣ ɩɚɪɚɥɥɟɥɶɧɨɣ ɨɫɢ Ɉɭ. ɉɨɷɬɨɦɭ ɜɫɟ ɫɟɦɟɣɫɬɜɨ ɧɚɤɥɨɧɧɵɯ ɩɪɹɦɵɯ ɞɨɩɨɥɧɹɟɬɫɹ ɩɪɹɦɵɦɢ
ɯ ɚ (4)
ɩɚɪɚɥɥɟɥɶɧɵɦɢ ɨɫɢ Ɉɭ ɍɪɚɜɧɟɧɢɟ ɩɨɥɭɱɟɧɨ ɢɡ ɭɪɚɜɧɟɧɢɹɩɪɢ ȼ = ɝɞɟ a C
A .
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ɍɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɢ
Ɇ1 |
ɉɭɫɬɶ ɞɚɧɵ ɬɨɱɤɢ Ɇ0(ɯ0 ɭ0 ɢ Ɇ1(ɯ1 ɭ1). |
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Ɍɪɟɛɭɟɬɫɹ ɧɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ |
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Ɇ0 |
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ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɷɬɢ ɬɨɱɤɢ Ⱦɥɹ ɪɟɲɟɧɢɹ |
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ɡɚɞɚɱɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɭɪɚɜɧɟɧɢɟɦ ȼ |
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S |
ɤɚɱɟɫɬɜɟ ɧɚɩɪɚɜɥɹɸɳɟɝɨ ɜɟɤɬɨɪɚ S ɜɨɫ |
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ɩɨɥɶɡɭɟɦɫɹ |
ɜɟɤɬɨɪɨɦ |
M 0M1 : |
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S M 0M1 (x1 x0;y1 y0 ) . |
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ɉɨɞɫɬɚɜɢɦ l = x1 – x0 |
ɢ m = y1 – y0 |
ɜ ɤɚɧɨɧɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ |
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ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɞɜɟ ɬɨɱɤɢ:
x x0 |
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y y0 |
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(5) |
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ɍɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɞɚɧɧɭɸ ɬɨɱɤɭ ɜ ɡɚɞɚɧ ɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ
ɉɭɫɬɶ ɞɚɧɚ ɬɨɱɤɚ Ɇ0(ɯ0 ɭ0 Ɍɪɟɛɭɟɬɫɹ ɧɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ Ɇ0 ɜ ɡɚɞɚɧɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ
Ɂɚɞɚɱɭ ɛɭɞɟɦ ɪɟɲɚɬɶ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɨɝɨ ɤɚɤ ɨɩɪɟɞɟɥɟɧɨ
ɧɚɩɪɚɜɥɟɧɢɟ ɩɪɹɦɨɣ ȿɫɥɢ ɧɚɩɪɚɜɥɟɧɢɟ ɡɚɞɚɟɬɫɹ ɜɟɤɬɨɪɨɦ S ɬɨ ɬɚɤɚɹ ɩɪɹɦɚɹ ɨɩɢɫɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ȿɫɥɢ ɡɚɞɚɧ ɭɝɥɨɜɨɣ ɤɨɷɮɮɢɰɢɟɧɬ k = k1 ɬɨ ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɛɭɞɟɬ ɧɚɯɨɞɢɬɶ ɜ ɮɨɪɦɟ y = k1x + b ɇɟɢɡɜɟɫɬɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ b ɧɚɣɞɟɦ ɢɡ ɭɫɥɨɜɢɹ y0 = k1x0 + b ɬɨɱɤɚ Ɇ0 ɩɪɢɧɚɞɥɟɠɢɬ ɩɪɹɦɨɣ ɇɚɣɞɟɧ ɧɨɟ b = y0 – k1x0 ɩɨɞɫɬɚɜɢɦ ɜ ɭɪɚɜɧɟɧɢɟ y = k1x + b ɂɫɤɨɦɨɟ ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ
y – y0 = k1(x – x0) . |
(6) |
ɉɪɢɦɟɪ. ɇɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱ ɤɭ Q ɩɚɪɚɥɥɟɥɶɧɨ ɩɪɹɦɨɣ ɯ-4ɭ+3=0.
Ɋɟɲɟɧɢɟ ɇɚɣɞɟɦ ɭɝɥɨɜɨɣ ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɹɦɨɣ ɯ –
4ɭ+ 3 = 0:
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Ⱦɥɹ ɢɫɤɨɦɨɣ ɩɪɹɦɨɣ ɭɝɥɨɜɨɣ ɤɨɷɮɮɢɰɢɟɧɬ ɛɭɞɟɬ ɬɚɤɢɦ ɠɟ ɬɚɤ ɤɚɤ ɩɪɹɦɵɟ ɩɚɪɚɥɥɟɥɶɧɵ ɉɨɞɫɬɚɜɢɦ ɞɚɧɧɵɟ ɜ ɭɪɚɜɧɟɧɢɟ
(6):
y 7 12 (x 2) .
4.4. ɍɝɨɥ ɦɟɠɞɭ ɞɜɭɦɹ ɩɪɹɦɵɦɢ
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n2 |
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Ⱦɚɧɵ ɭɪɚɜɧɟɧɢɹ ɞɜɭɯ ɩɪɹɦɵɯ |
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n1 |
Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɭɝɨɥ ɦɟɠɞɭ ɧɢɦɢ |
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ɍɝɨɥ ɦɟɠɞɭ ɞɜɭɦɹ ɧɟɩɚɪɚɥɥɟɥɶɧɵɦɢ |
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ɩɪɹɦɵɦɢ Į1 ɢ Į2 ɧɚɣɞɟɦ ɤɚɤ ɭɝɨɥ |
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ɦɟɠɞɭ ɧɚɩɪɚɜɥɹɸɳɢɦɢ ɜɟɤɬɨɪɚɦɢ
ɷɬɢɯ ɩɪɹɦɵɯ ɩɪɢ ɡɚɞɚɧɢɢ ɤɚɧɨɧɢɱɟ ɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɞɥɹ Į1 ɢ Į2 ɢɥɢ ɠɟ ɤɚɤ ɭɝɨɥ ɦɟɠɞɭ ɢɯ ɧɨɪɦɚɥɹɦɢ ɟɫɥɢ ɡɚɞɚɧɵ ɨɛɳɢɟ ɭɪɚɜɧɟɧɢɹ ɩɪɹɦɵɯ Į1 ɢ Į2.
ɉɭɫɬɶ ɡɚɞɚɧɵ ɞɜɟ ɩɪɹɦɵɟ Į1: |
x x1 |
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x x2 |
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ɇɚɩɪɚɜɥɹɸɳɢɟ ɜɟɤɬɨɪɵ ɷɬɢɯ ɩɪɹɦɵɯ S1 (l1,m1)ɢ
S2 (l2,m2 ).
ɍɝɨɥ ij ɦɟɠɞɭ ɩɪɹɦɵɦɢ ɧɚɣɞɟɦ ɢɡ ɫɤɚɥɹɪɧɨɝɨ ɩɪɨɢɡɜɟɞɟɧɢɹ
ɜɟɤɬɨɪɨɜ S1 ɢ S2 : cos |
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l1 l2 m1 m2 |
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S |
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l 2 m2 |
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ɉɭɫɬɶ ɡɚɞɚɧɵ ɨɛɳɢɟ ɭɪɚɜɧɟɧɢɹ ɩɪɹɦɵɯ 1 |
ɢ 2 : Ⱥ1ɯ ȼ1ɭ ɋ1 |
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ɢ Ⱥ2ɯ ȼ2ɭ ɋ2 Ɍɨɝɞɚ ɧɨɪɦɚɥɢ ɤ ɷɬɢɦ ɩɪɹɦɵɦ n1 |
(A1;B1) |
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ɢ n2 (A2;B2 ) ɢ cos |
n1 n2 |
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B1 B2 |
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ȿɫɥɢ |
1 : |
y k1x b1, |
2 : |
y k2 x b2 |
ɬɨ ɢɡ |
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k1x y b1 |
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k2 x y b 0 |
ɫɥɟɞɭɟɬ |
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cos |
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ɍɫɥɨɜɢɟɦ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɫɬɢ ɩɪɹɦɵɯ ɛɭɞɟɬ ɫɨɨɬɜɟɬɫɬɜɟɧ |
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ɧɨ |
0, ɥɢɛɨ A1A2 B1B2 0, ɥɢɛɨ k1k2 1 0. |
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l1l2 m1m2 |
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ɂɡ ɩɨɫɥɟɞɧɟɝɨ ɪɚɜɟɧɫɬɜɚ ɫɥɟɞɭɟɬ ɱɬɨ k2 1
k1 Ɍɚɤɢɦ
ɨɛɪɚɡɨɦ ɭɝɥɨɜɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɞɜɭɯ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɩɪɹ ɦɵɯ ɨɛɪɚɬɧɵɟ ɩɨ ɜɟɥɢɱɢɧɟ ɢ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵ ɩɨ ɡɧɚɤɭ
ɍɫɥɨɜɢɟɦ ɩɚɪɚɥɥɟɥɶɧɨɫɬɢ ɩɪɹɦɵɯ ɛɭɞɟɬ ɫɨɨɬɜɟɬɫɬɜɨɜɚɬɶ
l1 |
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A1 |
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4 Ɋɚɫɫɬɨɹɧɢɟ ɨɬ ɬɨɱɤɢ ɞɨ ɩɪɹɦɨɣ
Ⱦɚɧɚ ɩɪɹɦɚɹ Ⱥɯ ȼɭ ɋ ɢ ɬɨɱɤɚ Q(ɯ1 ɭ1 Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɬɨɱɤɢ Q ɞɨ ɩɪɹɦɨɣ ɗɬɨ ɪɚɫɫɬɨɹɧɢɟ ɧɚɯɨɞɢɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
h |
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Ax1 By1 C |
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(7) |
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A2 B2 |
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ɉɪɢɦɟɪ. Ⱦɚɧɵ ɜɟɪɲɢɧɵ ɬɪɟɭɝɨɥɶɧɢɤɚ Ⱥȼɋ: Ⱥ(–2,3), ȼ(1,12), ɋ ɇɚɣɬɢ ɭɪɚɜɧɟɧɢɟ ɫɬɨɪɨɧɵ Ⱥȼ ɭɪɚɜɧɟɧɢɟ ɜɵɫɨɬɵ ɋ' ɨɩɭɳɟɧɧɨɣ ɢɡ ɜɟɪɲɢɧɵ ɋ ɧɚ ɫɬɨɪɨɧɭ Ⱥȼ ɭɪɚɜɧɟɧɢɟ ɦɟ ɞɢɚɧɵ Ⱥȿ ɭɪɚɜɧɟɧɢɟ ɨɤɪɭɠɧɨɫɬɢ ɞɥɹ ɤɨɬɨɪɨɣ ɦɟɞɢɚɧɚ Ⱥȿ ɫɥɭɠɢɬ ɞɢɚɦɟɬɪɨɦ
Ɋɟɲɟɧɢɟ
ɍɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ Ⱥ(ɯ1, ɭ1 ɢ ȼ(ɯ2,
ɭ2 ɢɦɟɟɬ ɜɢɞ |
y y1 |
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ɑɬɨɛɵ ɧɚɣɬɢ ɭɪɚɜɧɟɧɢɟ ɫɬɨɪɨ |
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ɧɵ Ⱥȼ ɩɨɞɫɬɚɜɢɦ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɟɤ Ⱥ ɢ ȼ ɜ ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ
54
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ȼɵɫɨɬɚ ɋ' ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ ɫɬɨɪɨɧɟ Ⱥȼ ɚ ɩɨɬɨɦɭ ɢɯ ɭɝ ɥɨɜɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ kCD ɢ kAB ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɭɫɥɨɜɢɸ
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ɂɡ ɭɪɚɜɧɟɧɢɹ ɩɪɹɦɨɣ Ⱥȼ ɫɥɟɞɭɟɬ ɱɬɨ kAB ɬɨɝɞɚ |
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ɇɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɞɚɧɧɭɸ ɬɨɱɤɭ ɜ ɞɚɧɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ y – y1 = k(x – x1) ɉɨɞɫɬɚɜɢɜ ɜ ɭɪɚɜɧɟɧɢɟ ɤɨ ɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɋ ɢ ɭɝɥɨɜɨɣ ɤɨɷɮɮɢɰɢɟɧɬ kCD ɩɨɥɭɱɢɦ ɢɫɤɨɦɨɟ ɭɪɚɜɧɟɧɢɟ ɜɵɫɨɬɵ ɋ':
y 6 1 |
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3y 18 x 11; |
x 3y 29 0(ɋ') |
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Ɉɩɪɟɞɟɥɢɦ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ȿ ɉɪɢɦɟɧɹɟɦ ɮɨɪɦɭɥɵ ɞɟ |
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ɥɟɧɢɹ ɨɬɪɟɡɤɚ ɩɨɩɨɥɚɦ |
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9, E(6,9). |
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ɉɨ ɬɨɱɤɚɦ Ⱥ ɢ ȿ ɩɨɫɬɪɨɢɦ ɭɪɚɜɧɟɧɢɟ ɦɟɞɢɚɧɵ Ⱥȿ:
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ɍɪɚɜɧɟɧɢɟ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɤɫɚ R ɫ ɰɟɧɬɪɨɦ ɜ ɬɨɱɤɟ Ʉ(ɚ E) ɢɦɟɟɬ ɜɢɞ ɯ – ɚ)2 + (ɭ – b)2 = R2.
Ɍɚɤ ɤɚɤ ɩɨ ɭɫɥɨɜɢɸ ɦɟɞɢɚɧɚ Ⱥȿ ɹɜɥɹɟɬɫɹ ɞɢɚɦɟɬɪɨɦ ɢɫɤɨɦɨɣ ɨɤɪɭɠɧɨɫɬɢ ɬɨ ɰɟɧɬɪ ɨɤɪɭɠɧɨɫɬɢ Ʉ ɞɟɥɢɬ ɨɬɪɟɡɨɤ Ⱥȿ ɩɨɩɨɥɚɦ
ɇɚɯɨɞɢɦ |
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ɑɬɨɛɵ ɧɚɣɬɢ ɪɚɞɢɭɫ R ɨɤɪɭɠɧɨɫɬɢ ɞɨɫɬɚɬɨɱɧɨ ɧɚɣɬɢ ɪɚɫɫɬɨ ɹɧɢɟ ɦɟɠɞɭ ɬɨɱɤɚɦɢ Ⱥ ɢ Ʉ ɂɡɜɟɫɬɧɨ ɱɬɨ ɪɚɫɫɬɨɹɧɢɟ d ɦɟɠɞɭ ɞɜɭɦɹ ɬɨɱɤɚɦɢ ɩɥɨɫɤɨɫɬɢ Ɇ1(ɯ1,ɭ1 ɢ Ɇ2(ɯ2,ɭ2 ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ
ɮɨɪɦɭɥɟ d (x2 x1)2 (y2 y1)2 ɉɨɞɫɬɚɜɢɜ ɤɨɨɪɞɢɧɚɬɵ
55
ɬɨɱɟɤ Ⱥ ɢ Ʉ ɩɨɥɭɱɚɟɦ AK 
(2 2)2 (6 3)2 5 ɬ ɟ R=5.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ (ɯ – 2)2 + (ɭ – 6)2 = 25 – ɢɫɤɨɦɨɟ ɭɪɚɜɧɟɧɢɟ ɨɤɪɭɠɧɨɫɬɢ
ɉɪɢɦɟɪ. ȼ ɬɪɟɭɝɨɥɶɧɢɤɟ ɫ ɜɟɪɲɢɧɚɦɢ Ⱥ(2,3), ȼ(–1,0), ɋ(4,1) ɧɚɣɬɢ ɞɥɢɧɭ ɜɵɫɨɬɵ ɨɩɭɳɟɧɧɨɣ ɢɡ ɜɟɪɲɢɧɵ Ⱥ ɧɚ ɫɬɨɪɨɧɭ ȼɋ.
Ɋɟɲɟɧɢɟ ɋɨɫɬɚɜɢɦ ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɫɬɨɪɨɧɭ ȼɋ ɩɨ ɮɨɪɦɭɥɟ
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ɇɚɣɞɟɦ ɞɥɢɧɭ ɜɵɫɨɬɵ Ⱥȿ ɩɨ ɮɨɪɦɭɥɟ
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4.6 ɍɪɚɜɧɟɧɢɟ ɤɪɢɜɨɣ ɧɚ ɩɥɨɫɤɨɫɬɢ
Ɉɩɪɟɞɟɥɟɧɢɟ Ʌɢɧɢɹ ɧɚ ɩɥɨɫɤɨɫɬɢ ɧɚɡɵɜɚɟɬɫɹ ɚɥɝɟɛɪɚɢɱɟ ɫɤɨɣ ɟɫɥɢ ɟɟ ɦɨɠɧɨ ɡɚɞɚɬɶ ɭɪɚɜɧɟɧɢɟɦ P(x,y ɝɞɟ P(x,y) - ɰɟ ɥɵɣ ɦɧɨɝɨɱɥɟɧ ɨɬ ɩɟɪɟɦɟɧɧɵɯ ɯ ɭ.
ɋɬɟɩɟɧɶ ɦɧɨɝɨɱɥɟɧɚ P(x,y ɧɚɡɵɜɚɟɬɫɹ ɩɨɪɹɞɤɨɦ ɥɢɧɢɢ ɉɪɢɦɟɪɨɦ ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɥɢɧɢɢ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɹɜɥɹɟɬɫɹ ɩɪɹɦɚɹ ɜ R2 ɡɚɞɚɧɧɚɹ ɭɪɚɜɧɟɧɢɟɦ Ⱥɯ ȼɭ ɋ ɩɪɢɦɟɪɨɦ ɥɢɧɢɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɹɜɥɹɟɬɫɹ ɨɤɪɭɠɧɨɫɬɶ ɯ-ɚ)2+(ɭ-b)2=R2 ɉɪɢ ɨɛ ɪɚɛɨɬɤɟ ɞɚɧɧɵɯ ɩɪɚɤɬɢɤɢ ɢ ɩɪɢ ɩɨɫɬɪɨɟɧɢɢ ɷɦɩɢɪɢɱɟɫɤɢɯ ɡɚɜɢ ɫɢɦɨɫɬɟɣ ɱɚɫɬɨ ɩɪɢɯɨɞɢɬɫɹ ɢɫɩɨɥɶɡɨɜɚɬɶ ɤɪɢɜɵɟ ɥɢɧɢɢ ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɢɫɩɨɥɶɡɭɸɬ ɚɥɝɟɛɪɚɢɱɟɫɤɢɟ ɥɢɧɢɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ
Ɉɛɳɟɟ ɭɪɚɜɧɟɧɢɟ ɥɢɧɢɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɢɦɟɟɬ ɜɢɞ
Ⱥɯ2+ȼɭ2+ɋɯɭ 'ɯ ȿɭ F=0.
ɑɬɨɛɵ ɷɬɭ ɥɢɧɢɸ ɢɡɨɛɪɚɡɢɬɶ ɜ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɭɪɚɜ ɧɟɧɢɟ ɩɪɢɜɨɞɹɬ ɤ ɤɚɧɨɧɢɱɟɫɤɨɦɭ ɜɢɞɭ ɩɭɬɟɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɤɨɨɪɞɢɧɚɬ ɩɨɜɨɪɨɬɚ ɢ ɩɚɪɚɥɥɟɥɶɧɨɝɨ ɩɟɪɟɧɨɫɚ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ Ɋɚɫɫɦɨɬɪɢɦ ɛɨɥɟɟ ɩɪɨɫɬɨɣ ɫɥɭɱɚɣ ɭɪɚɜɧɟɧɢɹ ɩɪɢ ɋ=0:
Ⱥɯ2+ȼɭ2+'ɯ ȿɭ F=0.
56
Ɉɧɨ ɦɨɠɟɬ ɛɵɬɶ ɩɪɢɜɟɞɟɧɨ ɤ ɤɚɧɨɧɢɱɟɫɤɨɦɭ ɜɢɞɭ ɬɨɥɶɤɨ ɩɚɪɚɥɥɟɥɶɧɵɦ ɩɟɪɟɧɨɫɨɦ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ
4.7 Ʉɪɢɜɵɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɡɚɞɚɧɧɵɟ ɜ ɤɚɧɨɧɢɱɟɫɤɨɦ ɜɢɞɟ
Ɉɤɪɭɠɧɨɫɬɶɸ ɧɚɡɵɜɚɟɬɫɹ ɝɟɨɦɟɬɪɢɱɟɫɤɨɟ ɦɟɫɬɨ ɬɨɱɟɤ ɪɚɜɧɨɭɞɚɥɟɧɧɵɯ ɨɬ ɨɞɧɨɣ ɮɢɤɫɢɪɨɜɚɧɧɨɣ ɬɨɱɤɢ ɧɚɡɵɜɚɟɦɨɣ ɰɟɧɬɪɨɦ ɨɤɪɭɠɧɨɫɬɢ Ɋɚɫɫɬɨɹɧɢɟ ɨɬ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ ɨɤɪɭɠɧɨɫɬɢ ɞɨ ɟɝɨ ɰɟɧɬɪɚ ɧɚɡɵɜɚɟɬɫɹ ɪɚɞɢɭɫɨɦ ɨɤɪɭɠɧɨɫɬɢ
Y
M(x;y) |
ȿɫɥɢ ɰɟɧɬɪ |
ɨɤɪɭɠɧɨɫɬɢ |
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ɜ |
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ɬɨ ɭɪɚɜɧɟɧɢɟ |
ɨɤɪɭɠɧɨɫɬɢ |
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x x0 2 y y0 2 R2 .
0
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ɗɥɥɢɩɫɨɦ ɧɚɡɵɜɚɟɬɫɹ ɝɟɨɦɟɬɪɢɱɟɫɤɨɟ ɦɟɫɬɨ ɬɨɱɟɤ ɫɭɦɦɚ ɪɚɫɫɬɨɹɧɢɣ ɤɨɬɨɪɵɯ ɞɨ ɞɜɭɯ ɮɢɤɫɢɪɨɜɚɧɧɵɯ ɬɨɱɟɤ ɧɚɡɵɜɚɟɦɵɯ ɮɨɤɭɫɚɦɢ ɟɫɬɶ ɜɟɥɢɱɢɧɚ ɩɨɫɬɨɹɧɧɚɹ
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɜɵɜɟɫɬɢ ɤɚɧɨɧɢɱɟɫɤɨɟɩɪɨɫɬɟɣɲɟɟ ɭɪɚɜɧɟɧɢɟ ɷɥɥɢɩɫɚ ɩɪɢɦɟɦ ɡɚ ɨɫɶ Ox ɩɪɹɦɭɸ ɫɨ ɟɞɢɧɹɸɳɭɸ ɮɨɤɭɫɵ F1 ɢ F2 ɉɭɫɬɶ ɩɪɢ ɷɬɨɦ ɮɨɤɭɫɵ ɛɭɞɭɬ ɫɢɦɦɟɬ ɪɢɱɧɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɬ ɟ ɛɭɞɭɬ ɢɦɟɬɶ ɤɨɨɪɞɢɧɚ ɬɵ
57
F1 c;0 ɢ F2 c;0 . Ɂɞɟɫɶ ɱɟɪɟɡ ɫ ɨɛɨɡɧɚɱɟɧɨ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɮɨɤɭɫɚɦɢ
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ɷɥɥɢɩɫɚ ɢɦɟɟɬ ɜɢɞ |
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ɚ=b ɩɨɥɭɱɚɟɬɫɹ ɨɤɪɭɠɧɨɫɬɶ |
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Ƚɢɩɟɪɛɨɥɚ ɡɚɞɚɟɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ |
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ɧɚɡɵɜɚɟɬɫɹ ɪɚɜɧɨɛɨɱɧɨɣ ɉɨɥɭɨɫɶ b ɜ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɢ ɩɨɥɭɨɫɶ ɚ ɜɨ ɜɬɨɪɨɦ ɫɥɭɱɚɟ ɧɚɡɵɜɚɸɬɫɹ ɦɧɢɦɵɦɢ ɬ ɤ ɥɢɧɢɹ ɧɟ ɩɟɪɟɫɟɤɚ ɟɬ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɫɢ
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ɉɚɪɚɛɨɥɚ ɡɚɞɚɟɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ y2 qx ɥɢɛɨ x2 |
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4.8 ɉɪɢɜɟɞɟɧɢɟ ɨɛɳɟɝɨ ɭɪɚɜɧɟɧɢɹ ɥɢɧɢɣ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɤ ɤɚɧɨɧɢɱɟɫɤɨɦɭ ɜɢɞɭ
Ɂɚɞɚɱɚ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ ɱɬɨɛɵ ɩɪɢ ɚɧɚɥɢɡɟ ɭɪɚɜɧɟ ɧɢɹ Ⱥɯ2+ȼɭ2+'ɯ ȿɭ F=0 ɨɩɪɟɞɟɥɢɬɶ ɜɢɞ ɥɢɧɢɢ ɢ ɩɪɟɨɛɪɚɡɨɜɚɬɶ ɟɟ ɜ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɋɞɟɥɚɟɦ ɷɬɨ ɜ ɞɜɚ ɷɬɚɩɚ
ȼɵɞɟɥɢɦ ɜ ɨɛɳɟɦ ɭɪɚɜɧɟɧɢɢ ɩɨɥɧɵɟ ɤɜɚɞɪɚɬɵ ɨɬɧɨ
ɫɢɬɟɥɶɧɨ ɯ ɢ ɭ ɟɫɥɢ A 0 |
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ɉɪɢ Ⱥ ɥɢɛɨ ȼ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɤɜɚɞɪɚɬ ɧɟ ɜɵɞɟɥɹ |
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Ɉɛɨɡɧɚɱɢɦ x/ |
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ɬɟɦ ɫɚɦɵɦ ɩɟɪɟɯɨɞ ɤ ɧɨɜɨɣ ɤɨɨɪɞɢɧɚɬɧɨɣ ɫɢɫɬɟɦɟ Ɉ¶ ɯ ɭ ɫ
ɧɚɱɚɥɨɦ ɤɨɨɪɞɢɧɚɬ ɜ ɬɨɱɤɟ O/ |
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ɧɨɜɵɟ ɤɨɨɪɞɢɧɚɬɵ ɯ/ ɢ ɭ/ ɪɚɜɧɵ |
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ȼ ɤɚɧɨɧɢɱɟɫɤɨɣ ɡɚɩɢɫɢ ɢɦɟɟɦ ɷɥɥɢɩɫ ɢɥɢ ɝɢɩɟɪɛɨɥɭ ɜ |
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ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɡɧɚɤɨɜ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɪɢ A 0 |
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ɢ ɩɚɪɚɛɨɥɭ ɩɪɢ Ⱥ ɢ ȼ Ʌɢɧɢɢ ɢɡɨɛɪɚɠɚɟɦ ɜ ɫɢɫɬɟɦɟ ɤɨɨɪ ɞɢɧɚɬ Ɉ ɯ¶ ɭ¶ .
ɉɪɢɦɟɪ ɉɨɫɬɪɨɢɬɶ ɤɪɢɜɭɸ x2+2x+2ɭ2-8ɭ-10=0
59
Ɋɟɲɟɧɢɟ
1. ȼɵɞɟɥɢɦ ɩɨɥɧɵɟ ɤɜɚɞɪɚɬɵ ɫ ɯ ɢ ɭ:
(ɯ2+2ɯ+1)+2(ɭ2-4ɭ+4)-1-8-10=0, (ɯ+1)2+2(ɭ-2)2-19=0
Ɉɫɭɳɟɫɬɜɢɦ ɩɚɪɚɥɥɟɥɶɧɵɣ ɩɟɪɟɧɨɫ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ ɩɨ ɮɨɪɦɭɥɚɦ ɯ+1=ɯ/, ɭ-2=ɭ/ ɜ ɧɨɜɨɟ ɧɚɱɚɥɨ Ɉ/(-1,2).
ȼ ɧɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɢɦɟɟɦ x/2+2ɭ/2-19=0
ɢɥɢ |
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ɜ ɤɚɧɨɧɢɱɟɫɤɨɦ ɜɢɞɟ |
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ɉɨɥɭɱɢɥɢ ɷɥɥɢɩɫ ɫ ɩɨɥɭɨɫɹɦɢ a |
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ɉɪɢɦɟɪ ɉɨɫɬɪɨɢɬɶ ɤɪɢɜɭɸ –8ɭ2+2ɭ+7ɯ-4=0
Ɋɟɲɟɧɢɟ.
ȼɵɞɟɥɢɦ ɩɨɥɧɵɣ ɤɜɚɞɪɚɬ ɫ ɭ:
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60