F4 A-MATHS.(4)

a)
AB / PB = 3,
(AP+PB) : PB = 3:1
AP : PB = 2:1

Px = (0×2+16×1)÷(2+1)
= 16/3
Py = (8×2+0×1)÷(2+1)
= 16/3
P的坐標為 (16/3, 16/3)

b)
By area formula,
When Cx is 0,
Area of △OAB = 8 × area of △CPB
8*16/2 = 8 ×
圖片參考：http://i169.photobucket.com/albums/u209/lautszki/07-05-05/A0.gif

±16 = 16/3 × Cy – 16/3 × 8
± 3 = Cy – 8
Cy = 11 or 5.

When Cy is 0,
Area of △OAB = 8 × area of △CPB
8*16/2 = 8 ×
圖片參考：http://i169.photobucket.com/albums/u209/lautszki/07-05-05/A1.gif

±16 = 8 ×Cx – 16/3 × 8 – 16/3 ×Cx
±16 + 16/3*8 = (8 – 16/3) Cx
176/3 = 8/3 Cx or 80/3 = 8/3Cx
Cx = 22 or 10.

C is at (0, 11), (0, 5), (22, 0) or (10,0).

a ) AB/PB=3
AB : PB=3 :1
AP : PB=2 :1
P點的x-坐標
=[2(0)+1(16)]/(2+1)
=16/3
P點的y-坐標
=[2(8)+1(0)]/(2+1)
=16/3
所以P點的坐標= (16/3 , 16/3)

b )三角形OAB的面積
=(1/2) (OA)(OB)
=(1/2)(16)(8)
=64
所以三角形CPB的面積=(1/8)(64) = 8

如果C在Y坐標軸上 , C的x-坐標 =0。 (與B 點成垂直線)
三角形CPB的面積=(1/2)(BC)(16/3)
(8)(2)(3/16)=BC
BC=3
所以C的y-坐標=5
所以C的坐標=(0,5)

如果C在X坐標軸上 , C的y-坐標 =0。
設C的x-坐標=a。
三角形CPB的面積
=(1/2) | (16/3)(a) + (16/3)(8) - 8a |
=| (64/3)-(8a/3) |
所以 ,
8 =| (64/3)-(8a/3) |
8 =(64/3)-(8a/3) or -8 =(64/3)-(8a/3)
24=64-8a or -24=64-8a
8a = 40 or 8a = 88
a = 5 or 11
所以所以C的坐標=(5,0) 或 (11,0)。