Amaths F.4

Let A(4 , 3) and B(2 , 8) be two given points. P is a point on AB such that AP : PB = r : 1,Find the ratio in which the line 4x － y－- 10 = 0 divides the line segment AB.
Sol
set P(a，b) is the intesection of AB and 4x－y－10=0
then
(4－a):(a－2)=(3－b):(b-8)
(4－a)(b－8)=(a－2)(3－b)
4b－32－ab+8a=3a－ab－6+2b
4b－32+8a=3a－6+2b
5a+2b=26
4a－b=10
8a－2b=20
13a=46
a=46/13
b=4a-10=54/13
ratio= (4－a):(a－2)
=(4－46/13):(46/13－2)
=(52－46):(46－26)
=6:20
=3:10

The x - coordinate of P is (2r + 4)/(1 + r)
The y - coordinates of P is ((8r + 3)/(1 + r).
Since P is on line 4x - y - 10 = 0. Therefore,
4(2r + 4)/(1 + r) - (8r + 3)/(1 + r) - 10 = 0
8r + 16 - 8r - 3 - 10 - 10r = 0
10r = 3
r = 3/10.

Equation of the line AB : ( y – 3 ) / ( x – 4 ) = ( 3 – 8 ) / ( 4 – 2 )
5x + 2y – 26 = 0


Let P(x,y) be the point of intersection of the line 4x - y - 10 = 0 and the line joining AB.
5x + 2y – 26 = 0 . . . . . (1)
4x - y - 10 = 0 . . . . . . . . (2)

(1) + 2 X (2): 13x – 46 = 0
x = 46 / 13

Since AP : PB = r : 1
46 / 13 = ( 4 + 2r ) / ( 1 + r )
r = 3 / 10