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Giải bài 3.7 trang 52 Chuyên đề học tập Toán 10 - Kết nối tri thức với cuộc sống

Trong mặt phẳng tọa độ Oxy, cho hypebol có phương trình chính tắc (frac{{{x^2}}}{9} - frac{{{y^2}}}{4} = 1)

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Đề bài

Trong mặt phẳng tọa độ Oxy, cho hypebol có phương trình chính tắc \(\frac{{{x^2}}}{9} - \frac{{{y^2}}}{4} = 1\) Xác định tọa độ các đỉnh, độ dài các trục, tâm sai và phương trình các đường chuẩn của hypebol

Phương pháp giải - Xem chi tiết

Cho hypebol \(\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1\) + 2 đỉnh: \({A_1}\left( { - a;0} \right),{A_2}\left( {a;0} \right),\) + Độ dài trục thực: 2a, độ dài trục ảo: 2b.  \(c = \sqrt {{a^2} + {b^2}} \) + Tâm sai của hypebol: \(e = \frac{c}{a}\) + Đường chuẩn: \({\Delta _1}:x =  - \frac{a}{e}\) và \({\Delta _2}:x = \frac{a}{e}\).

Lời giải chi tiết

Ta có phương trình chính tắc của hypebol là: \(\frac{{{x^2}}}{9} - \frac{{{y^2}}}{4} = 1\).\( \Rightarrow a = 3,b = 2,c = \sqrt {{a^2} + {b^2}}  = \sqrt {13} \)+ 2 đỉnh: \({A_1}\left( { - 3;0} \right),{A_2}\left( {3;0} \right),\)+ Độ dài trục thực: 2a = 6, độ dài trục ảo: 2b = 4.+ Tâm sai của hypebol: \(e = \frac{{\sqrt {13} }}{3}\)+ Đường chuẩn: \({\Delta _1}:x =  - \frac{3}{{\frac{{\sqrt {13} }}{3}}} \Leftrightarrow x =  - \frac{{9\sqrt {13} }}{{13}}\) và \({\Delta _2}:x = \frac{{9\sqrt {13} }}{{13}}\).

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