[칼?럼] 2022개정 기하 주요 개념 복습

Question

Noi dung chi tiet mon Hinh hoc Chuong trinh giao duc cai cach nam 2022 cua Han Quoc

Chuong 1. Duong Conic

Parabol
Parabol la tap hop cac diem cach deu mot diem co dinh goi la tieu diem F(p, 0) va mot duong thang co dinh goi la duong chuan x = -p.
Phuong trinh chinh tac co dang y^2 = 4px (voi p khac 0) hoac x^2 = 4py.
Phuong trinh tiep tuyen cua parabol y^2 = 4px tai diem M(x1, y1) nam tren parabol la y1 * y = 2p * (x + x1). Phuong trinh tiep tuyen co he so goc m la y = mx + p/m (voi m khac 0).
Elip
Elip la tap hop cac diem co tong khoang cach den hai diem co dinh, goi la hai tieu diem F(c, 0) va F'(-c, 0), bang mot hang so khong doi 2a (voi a lon hon c va c^2 = a^2 - b^2).
Phuong trinh chinh tac co dang (x^2 / a^2) + (y^2 / b^2) = 1 (voi a lon hon b va b lon hon 0).
Phuong trinh tiep tuyen cua elip tai diem M(x1, y1) nam tren elip la (x1 * x / a^2) + (y1 * y / b^2) = 1. Phuong trinh tiep tuyen co he so goc m la y = mx +- can bac hai cua (a^2 * m^2 + b^2).
Hypebol
Hypebol la tap hop cac diem co hieu khoang cach tuyet doi den hai diem co dinh, goi la hai tieu diem F(c, 0) va F'(-c, 0), bang mot hang so khong doi 2a (voi c lon hon a va c^2 = a^2 + b^2).
Phuong trinh chinh tac co dang (x^2 / a^2) - (y^2 / b^2) = 1 (voi a lon hon 0, b lon hon 0) va co cac duong tiem can la y = (b/a) * x va y = -(b/a) * x.
Phuong trinh tiep tuyen cua hypebol tai diem M(x1, y1) nam tren hypebol la (x1 * x / a^2) - (y1 * y / b^2) = 1. Phuong trinh tiep tuyen co he so goc m la y = mx +- can bac hai cua (a^2 * m^2 - b^2).

Chuong 2. Vector trong mat phang

Cac phep toan vector
Vector la mot doan thang co huong, duoc xac dinh boi diem dau va diem cuoi. Do dai cua vector a duoc ky hieu la |a|.
Phep cong hai vector duoc thuc hien theo quy tac ba diem: vector AB + vector BC = vector AC, hoac quy tac hinh binh hanh.
Phep nhan vector a voi mot so thuc k tao ra vector ka cung huong voi vector a neu k lon hon 0, va nguoc huong neu k nho hon 0. Do dai cua vector ka bang tri tuyet doi cua k nhan voi do dai vector a.
Toa do va tich vo huong cua vector phang
Neu bieu dien vector a = (a1, a2) va vector b = (b1, b2) trong he toa do Descartes, ta co phep cong vector a + vector b = (a1 + b1, a2 + b2) va phep nhan ka = (k * a1, k * a2).
Tich vo huong cua hai vector a va vector b duoc dinh nghia la tich do dai cua chung nhan voi cosin cua goc xen giua: a . b = |a| * |b| * cos(theta). Bieu dien qua toa do: a . b = a1 * b1 + a2 * b2.
Neu hai vector vuong goc voi nhau thi tich vo huong cua chung bang 0, tuc la a1 * b1 + a2 * b2 = 0.

Chuong 3. Hinh hoc khong gian va Toa do khong gian

Hinh hoc khong gian
Xac dinh vi tri tuong doi giua duong thang va duong thang, duong thang va mat phang, mat phang va mat phang trong khong gian ba chieu.
Dinh ly ba duong vuong goc: Neu duong thang a vuong goc voi mat phang alpha tai diem o, duong thang b nam trong alpha khong di qua o, va duong thang phan chieu cua a tren alpha vuong goc voi b, thi duong thang a cung vuong goc voi b.
Hinh chieu vuong goc cua mot diem, mot doan thang hoac mot da giac len mot mat phang. Dien tich hinh chieu S' duoc tinh bang cong thuc S' = S * cos(theta), voi S la dien tich goc va theta la goc giua hai mat phang.
Toa do khong gian va vector trong khong gian
Mot diem M trong khong gian ba chieu duoc xac dinh boi bo ba so (x, y, z). Khoang cach giua hai diem A(x1, y1, z1) va B(x2, y2, z2) duoc tinh bang cong thuc d = can bac hai cua ((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2).
Toa do diem chia trong doan thang AB theo ti le m : n la ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n), (mz2 + nz1)/(m+n)).
Phuong trinh mat cau co tam I(a, b, c) va ban kinh R la (x - a)^2 + (y - b)^2 + (z - c)^2 = R^2.
Vector trong khong gian co ba thanh phan a = (a1, a2, a3). Cac phep toan cong, tru, nhan voi mot so va tich vo huong a . b = a1 * b1 + a2 * b2 + a3 * b3 deu duoc thuc hien tuong tu nhu trong mat phang nhung co them thanh phan toa do thu ba z.

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