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Giải bài 1 trang 70 SGK Toán 10 tập 2 – Chân trời sáng tạo

Viết phương trình chính tắc của: a) Elip có trục lớn bằng 20 và trục nhỏ bằng 16

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Đề bài

Viết phương trình chính tắc của: a) Elip có trục lớn bằng 20 và trục nhỏ bằng 16 b) Hypebol có tiêu cự \(2c = 20\) và độ dài trục thực \(2a = 12\) c) Parabol có tiêu điểm \(F\left( {\frac{1}{2};0} \right)\)

Phương pháp giải - Xem chi tiết

a)       Bước 1: Từ giải thiết xác định a, b, c

Bước 2: Phương trình chính tắc của elip có dạng \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\) với \(M(x;y) \in (E);b = \sqrt {{a^2} - {c^2}} \) b) Phương trình chính tắc của hypebol có dạng \(\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1\) với \(M(x;y) \in (H);b = \sqrt {{c^2} - {a^2}} \) c) Phương trình chính tắc của parabol có dạng \({y^2} = 2px\) với \(M(x;y) \in 

Lời giải chi tiết

a) Ta có \(2a = 20 \Rightarrow a = 10,2b = 16 \Rightarrow b = 8\).Vậy phương trình chính tắc của elip có dạng \(\frac{{{x^2}}}{{100}} + \frac{{{y^2}}}{{64}} = 1\)b) Ta có \(2a = 12 \Rightarrow a = 6,2c = 20 \Rightarrow c = 10\), suy ra \(b = \sqrt {{c^2} - {a^2}}  = \sqrt {{{10}^2} - {6^2}}  = 8\)Vậy phương trình chính tắc của hypebol có dạng \(\frac{{{x^2}}}{{36}} - \frac{{{y^2}}}{{64}} = 1\)c) Ta có tiêu điểm \(F\left( {\frac{1}{2};0} \right)\).Do đó, \(\frac{p}{2} = \frac{1}{2}\) suy ra \(p = 1\).Vậy phương trình chính tắc của parabol là \({y^2} = 2x\).

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