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Giải bài 2 trang 64 Chuyên đề học tập Toán 10 – Chân trời sáng tạoChuyên đề học tập Toán 10 – Chân trời sáng tạo

Viết phương trình của đường conic có tâm sai bằng 1, tiêu điểm (F(1;0)) và đường chuẩn là (Delta :x + 1 = 0)

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Đề bài

Viết phương trình của đường conic có tâm sai bằng 1, tiêu điểm \(F(1;0)\) và đường chuẩn là \(\Delta :x + 1 = 0\)

Phương pháp giải - Xem chi tiết

Bước 1: Xác định loại đường conic dựa vào tâm sai e: + \(0 < e < 1\) thì conic là đường elip + \(e = 1\) thì conic là đường parabol + \(e > 1\) thì conic là đường hypebol Bước 2: Tìm tập hợp các điểm M sao cho \(\frac{{MF}}{{d(M,\Delta )}} = e\) Từ đó kết luận phương trình đường conic.

Lời giải chi tiết

Đường conic có tâm sai bằng 1 thì là parabol.Điểm \(M(x,y)\) thuộc đường conic khi và chỉ khi\(\begin{array}{l}\frac{{MF}}{{d(M,\Delta )}} = 1 \Leftrightarrow \frac{{\sqrt {{{\left( {x - 1} \right)}^2} + {y^2}} }}{{\left| {x + 1} \right|}} = 1\\ \Leftrightarrow \sqrt {{{\left( {x - 1} \right)}^2} + {y^2}}  = \left| {x + 1} \right|\\ \Leftrightarrow {\left( {x - 1} \right)^2} + {y^2} = {\left( {x + 1} \right)^2}\\ \Leftrightarrow {y^2} = 4x\end{array}\)Vậy phương trình đường parabol là \({y^2} = 4x\)

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