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TG
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TT
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H24
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H24
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NN
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NL
27 tháng 2 2023 lúc 22:45

Đề chính xác là \(\left|\overrightarrow{EB}-3\overrightarrow{EC}\right|\) đạt min đúng ko?

\(\overrightarrow{AB}=\left(2;-4\right)=2\left(1;-2\right)\) nên đường thẳng AB nhận \(\left(2;1\right)\) là 1 vtpt

Phương trình AB:

\(2\left(x+2\right)+1\left(y-1\right)=0\Leftrightarrow2x+y+3=0\)

Do E thuộc AB, đặt \(E\left(a;b\right)\Rightarrow2a+b+3=0\Rightarrow b=-2a-3\)

\(\Rightarrow E\left(a;-2a-3\right)\) \(\Rightarrow\left\{{}\begin{matrix}\overrightarrow{EB}=\left(-a;2a\right)\\\overrightarrow{EC}=\left(1-a;2a+4\right)\end{matrix}\right.\)

\(\Rightarrow\overrightarrow{EB}-3\overrightarrow{EC}=\left(2a-3;-4a-12\right)\)

\(\Rightarrow\left|\overrightarrow{EB}-3\overrightarrow{EC}\right|=\sqrt{\left(2a-3\right)^2+\left(-4a-12\right)^2}=\sqrt{20a^2+84a+153}\)

\(=\sqrt{20\left(a+\dfrac{21}{10}\right)^2+\dfrac{324}{5}}\ge\sqrt{\dfrac{324}{5}}\)

Dấu = xảy ra khi \(a+\dfrac{21}{10}=0\Rightarrow a=-\dfrac{21}{10}\)

\(\Rightarrow E\left(-\dfrac{21}{10};\dfrac{6}{5}\right)\)

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NN
27 tháng 2 2023 lúc 22:07

|(EB) ⃗ |+| 3(EC) ⃗ |

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NL
27 tháng 2 2023 lúc 22:08

Lỗi để rồi em

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HN
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KT
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NT
30 tháng 12 2021 lúc 21:36

\(\overrightarrow{CA}=\left(1-x_C;-2\right)\)

\(\overrightarrow{CB}=\left(-2-x_C;2\right)\)

\(\Leftrightarrow\left(x_C-1\right)\left(x_C+2\right)-4=0\)

\(\Leftrightarrow x_C^2+x_C-6=0\)

hay \(x_C=-3\)

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H24
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NL
15 tháng 2 2022 lúc 22:22

Áp dụng công thức trọng tâm:

\(\left\{{}\begin{matrix}x_C=3x_G-x_A-x_B=-3\\y_C=3y_G-y_A-y_B=-4\end{matrix}\right.\) \(\Rightarrow C\left(-3;-4\right)\)

\(\Rightarrow\overrightarrow{CA}=\left(4;5\right)\) ; \(\overrightarrow{AB}=\left(1;2\right)\)

Đường cao d đi qua B vuông góc AC nên nhận \(\overrightarrow{CA}=\left(4;5\right)\) là 1 vtpt

Phương trình d: 

\(4\left(x-2\right)+5\left(y-3\right)=0\Leftrightarrow4x+5y-23=0\)

Đường cao d1 đi qua C vuông góc AB nên nhận (1;2) là 1 vtpt

Phương trình d1:

\(1\left(x+3\right)+2\left(y+4\right)=0\Leftrightarrow x+2y+11=0\)

H là giao điểm d và d1 nên tọa độ thỏa mãn:

\(\left\{{}\begin{matrix}4x+5y-23=0\\x+2y+11=0\end{matrix}\right.\)  \(\Rightarrow\left\{{}\begin{matrix}x=\dfrac{101}{3}\\y=-\dfrac{67}{3}\end{matrix}\right.\) \(\Rightarrow H\left(\dfrac{101}{3};-\dfrac{67}{3}\right)\)

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H24
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TD
15 tháng 6 2022 lúc 20:31

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