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Giải bài 5 trang 48 Chuyên đề học tập Toán 10 – Chân trời sáng tạo

Trái Đất chuyển động theo một quỹ đạo là đường elip có tâm sai là 0,0167 và nhận tâm Mặt trời là một tiêu điểm

﷽Tổng hợp đề thi học kì 2 lớp 10 tất cả các môn - Chân trời sáng tạo

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Đề bài

Trái Đất chuyển động theo một quỹ đạo là đường elip có tâm sai là 0,0167 và nhận tâm Mặt trời là một tiêu điểm. Cho biết khoảng cách gần nhất giữa Trái Đất và tâm Mặt Trời là khoảng 147 triệu km, tính khoảng cách xa nhất giữa Trái Đất và tâm Mặt Trời. (Nguồn: )

Phương pháp giải - Xem chi tiết

Cho PTCT: \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\), \({F_1}( - c;0)\) là một tiêu điểm. + Tâm sai của elip: \(e = \frac{c}{a}\) Điểm M bất kì thuộc elip, khi đó: \(M{F_1}\) nhỏ nhất bằng \(a - c\) khi M trùng \({A_1}( - a;0)\) \(M{F_1}\) lớn nhất bằng \(a + c\) khi M trùng \({A_2}(a;0)\)

Lời giải chi tiết

Gọi PTCT của quỹ đạo hình elip đó là: \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\),+ Tâm sai của elip: \(e = \frac{c}{a} = 0,0167\) (1)Giả sử Mặt trời là tiêu điểm \({F_1}( - c;0)\).Điểm M bất kì thuộc elip là vị trí của Trái Đất trong quỹ đạo, khi đó:\(M{F_1}\) nhỏ nhất bằng \(a - c = 147\) (2)Từ (1) và (2) suy ra \(a \approx 149,5;c \approx 2,5\)\(M{F_1}\) lớn nhất bằng \(a + c = 149,5 + 2,5 = 152\)Vậy khoảng cách xa nhất giữa Trái Đất và tâm Mặt Trời là \(152\) triệu km. 

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