D B1 OB^ CH ± OBC HO =90 "
d) Gpi Rla bdn kinh dirdng tr6n tiep xuc dong thdi vdi (C,), (C2) va BC Hay tinh R theo R, va R2.
Hvldng d i n giai
Bai 1:
a) Va + b - c = yfa + \/h - yfc o Va + b - c + N/C = >/a + >/b 2 2 2 2
o (Va + b - c + Vc) = (Va + Vb)
o a + b - c + c + 2Vc(a + b - c ) = a + b + 2Vab
o Vc(a + b - c) - Vab o c(a + b - c) = ab o c(a + b - c) - ab = 0 o c a + c(b - c) - ab = 0 o c(b - c) - ăb - c) = 0 o (b - c)(c - a) = 0 o c a + c(b - c) - ab = 0 o c(b - c) - ăb - c) = 0 o (b - c)(c - a) = 0
"(a + b - c ) ' = a 2 + b 2 - c M = â) (a + b - c) ' = a 2 + b 2 - c M = b ^ ) ' Vay (a + b - c)^ = â + b^ -
Dieu ngiTdc lai cung dung. That vay :
(a + b - c)^ = â + b^ - c^ o â + b^ + + 2ab - 2ca - 2bc = a H b^ - c^ o 2 c ^ + 2ab - 2ca - 2bc = 0 o be - c^ - ab + ca = 0 'b - c = 0 "b = c <=> c - a = 0 c = a o c ( b - c) - ăb - c) = 0 o (b - c)(c - a) = 0 <:> " b - c = 0 c - a = 0 b = c c = a ' V ^ T b ^ = V I + Vb - V^(= Vay V I T b : : ^ = V I + V b - V ^ Va + b - c = Va + Vb - Vc(= Vb) Cty TNHt) M ! V l;VVll Ki b) A = b + c - 16abc. A p dung bat d^ng thiJc C6-si cho hai so diTdng, ta c6 :
1 = (a + b + c)^ > 4ăb + c) nen b + c > 4ăb + c)^ , Ma (b + c)^ > 4bc. Do do b + c > 16abc =J> b + c - 16abc > 0
A > 0 phúdng trinh luon cd nghiem.
Nhan xet: " "
a) De nhan ra tur Va + b - c = Va + Vb - Vc se cd diTdc b = c hoSc c = ạ Do vay cd (a + b - c)^ = â + b^ - c l Do vay cd (a + b - c)^ = â + b^ - c l
b) ChiJng minh b + c - 16abc > 0 neu cd a, b, c di/dng thoa man a + b + c = 11^ qua quen thuoc. Bat d^ng thiirc (x + y)^ > 4xy giup den vdi Idi giai bai loan. qua quen thuoc. Bat d^ng thiirc (x + y)^ > 4xy giup den vdi Idi giai bai loan.
Bai 2: T ị y - / . . ^
a) T a c d : a ' - a = ăá*- l ) = ăâ- l)(â + 1) = ăâ- l ) ( a ^ - 4 + 5) v = ăâ - l)(â - 4) + 5ăá - 1) = ăa + l)(a - l)(a + 2)(a - 2) +5ăâ - 1) ! 5 V i a - 2 ; ^ - 1 ; a ; a + I ;a + 2 i a 5 s o nguyen lien tiep nen cd mot s6' chia het cho 5 ăa + l)(a - l)(a + 2)(a - 2) i 5.
Mat khac 5ăâ - 1 ) 1 5 . Ttfdng tvT cd : b ' - b i 5 ; c* - c i 5
Ma a + b + c = 0 do đ â + b ' + c^ = (â - a) + (b^ - b) + (c^ - c) ! 5 b) x H y^ = xy + X + y o 2x^ + 2y^ = 2xy + 2x + 2y o ( x + y ' - 2xy) + ( x ^ - 2x + 1) + ( y ^ - 2y + 1) = 2 o ( x - y ) ^ + ( x - l)^ + ( y - 1)^ = 2 ( 2 = 0^+ 1^+ 1^) ( x - y ) ' = l ^ ( x - l ) ' = l ^ ( y - l f = 0 ^ ^ _ ( x - y ) ' : . l ^ ( x - l ) ' = 0 ^ ( y - l) ' = l 2 X = 0; y = 0 x = 2; y- 2 X =:0;y = 1 X = 2; y = r X = l; y = 0 [ x = 1; y = 2 Cdc cap so nguyen (x ; y) thoa man phi/dng trinh la :
( 0 ; 0 ) ; ( 2 ; 2 ) ; ( 0; I) ; ( 2 ; 1 ) ; ( 1 ; 0 ) ; ( 1 ; 2 )
Nhfinxet:
| » ) V d i moi X e Z, de dang chtfng minh diTdc x ' - x ! 5. Chung ta da cd Idi giai bai todn va hdn the nffa c6n nhan ra r^ng c6n cd á + b ' + c^ : 30.