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Giải bài 1 trang 10 Chuyên đề học tập Toán 11 Chân trời sáng tạo

Cho đường thẳng d đi qua tâm O của đường tròn (C) và cắt (C) tại A và B.

🥂Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Chân trời sáng tạo

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

🎐Cho đường thẳng d đi qua tâm O của đường tròn (C) và cắt (C) tại A và B. Tìm ảnh của đường tròn (C) qua phép chiếu vuông góc lên d.

Phương pháp giải - Xem chi tiết

Tìm ảnh của một số điểm bất kì trên đường tròn (C) qua phép chiếu vuông góc lên d. Từ đó rút ra kết luận.

Lời giải chi tiết

Ta đặt f là phép chiếu vuông góc lên d.Vì A, B là giao điểm của đường thẳng d và đường tròn (C) nên \(A{\rm{ }} = {\rm{ }}f\left( A \right),{\rm{ }}B{\rm{ }} = {\rm{ }}f\left( B \right){\rm{ }}\left( 1 \right)\)Lấy điểm M ∈ (C) sao cho M ≠ A và M ≠ B.Kẻ \(MM' \bot d\)  tại M’.Khi đó ta có \(M' = {\rm{ }}f\left( M \right).\)Mà AB là đường kính của đường tròn (C) nên M’ nằm trên đoạn thẳng AB.Tương tự như vậy, mỗi điểm N bất kì di động trên đường tròn (C) sao cho N ≠ A và N ≠ B thì ảnh N’ của N qua f đều nằm trên đoạn thẳng AB (2)Từ (1), (2), ta thu được ảnh của đường tròn (C) qua phép chiếu vuông góc lên d là đoạn thẳng AB hay \(f\left( {\left( C \right)} \right){\rm{ }} = {\rm{ }}AB.\)

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