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Bài 7.22 trang 50 SGK Toán 11 tập 2 - Cùng khám phá

Cho parabol \(\left( P \right):y = {x^2} + 2x + 3\).
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Đề bài

Cho parabol \(\left( P \right):y = {x^2} + 2x + 3\). Gọi \(\Delta \) là tiếp tuyến của \(\left( P \right)\) tại điểm \(M\) và có hệ số góc bằng \( - 2\). Phương trình của \(\Delta \) là A. \(y =  - 2x + 3.\) B. \(y = 2x - 3.\) C. \(y = 2x + 3.\) D. \(y =  - 2x - 3.\)

Phương pháp giải - Xem chi tiết

Từ \(f'\left( {{x_0}} \right) = k\) giải phương trình tìm được \({x_0}\) Từ đó tìm được điểm \({M_0}\left( {{x_0};{y_0}} \right)\) với \({y_0} = f\left( {{x_0}} \right)\) PTTT tại điểm \({M_0}\left( {{x_0};{y_0}} \right)\) là \(y = f'\left( {{x_0}} \right).\left( {x - {x_0}} \right) + {y_0}\)

Lời giải chi tiết

Gọi \({M_0}\left( {{x_0};{y_0}} \right)\) là tọa độ của tiếp điểmTa có \(y' = 2x + 2\)Vì hệ số góc \(k =  - 2\)  nên ta có \(y'\left( {{x_0}} \right) =  - 2 \Leftrightarrow 2{x_0} + 2 =  - 2 \Leftrightarrow {x_0} =  - 2\)\( \Rightarrow {y_0} = y\left( { - 2} \right) = {\left( { - 2} \right)^2} + 2.\left( { - 2} \right) + 3 = 3\)Vậy phương trình tiếp tuyến cần tìm là \(y = f'\left( {{x_0}} \right).\left( {x - {x_0}} \right) + {y_0}\)\( \Leftrightarrow y =  - 2\left( {x + 2} \right) + 3 \Leftrightarrow y =  - 2x - 1\)

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