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Giải bài 4.23 trang 70 SGK Toán 10 – Kết nối tri thức

Trong mặt phẳng tọa độ Oxy, cho hai điểm A (1; 2), B(-4; 3). Gọi M (t; 0) là một điểm thuộc trục hoành. a) Tính AM.BM theo t. b) Tính t để góc AMB = 90^o

Tổng hợp đề thi học kì 2ꦡ lớp 10 tất cả các môn - Kết nối tri thức

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Đề bài

Trong mặt phẳng tọa độ Oxy, cho hai điểm A (1; 2), B(-4; 3). Gọi M (t; 0) là một điểm thuộc trục hoành. a) Tính \(\overrightarrow {AM} .\overrightarrow {BM} \) theo t. b) Tính t để \(\widehat {AMB} = {90^o}\)

Phương pháp giải - Xem chi tiết

+) Nếu vecto \(\overrightarrow {AM} (x;y)\) và \(\overrightarrow {BM} (a;b)\) thì \(\overrightarrow {AM} .\overrightarrow {BM}  = xa + yb\) +) \(\widehat {AMB} = {90^o} \Leftrightarrow AM \bot BM\)

Lời giải chi tiết

 

a)  Ta có: A (1; 2), B(-4; 3) và M (t; 0).

\(\begin{array}{l}
\Rightarrow \overrightarrow {AM} = (t - 1; - 2),\;\overrightarrow {BM} = (t + 4; - 3)\\
\Rightarrow \overrightarrow {AM} .\overrightarrow {BM} = (t - 1)(t + 4) + ( - 2)( - 3)\\
\quad \quad \quad \quad \quad \quad= {t^2} + 3t + 2.
\end{array}\)

b) Để \(\widehat {AMB} = {90^o}\) hay \(AM \bot BM\) thì \(\overrightarrow {AM} .\overrightarrow {BM}  = 0\) \(\begin{array}{l} \Leftrightarrow {t^2} + 3t + 2 = 0\\ \Leftrightarrow \left[ \begin{array}{l}t =  - 1\\t =  - 2\end{array} \right.\end{array}\) Vậy t = -1 hoặc t = -2 thì \(\widehat {AMB} = {90^o}\).

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