In what ratio is the line joining (1,2) (4,3) divided by the line joining (2,3) (4,1)??

Here lines AB and CD intersect each  other at point O as shown below:

Slope of the line AB = 1-34-2=-22=-1
Equation of line AB passing through the points A(2, 3) having the slope -1 is;
y=-x+5......(i)
Also slope of the line CD = 3-24-1=13
Equation of line CD passing through the points C(1, 2) having the slope 13 is;
3y=x+5....(ii)
Now solving equation (i) and (ii) we have;
3y+y=x+5-x+54y=10y=52
then, x=52
Therefore both lines intersect at point O52,52.
Now the distance between the point C1,2 and O52,52= CO = 52-22+52-12=102
Also the distance between the point D4,3 and O52,52= DO = 52-32+52-42=102
So required ratio = CODO=10/210/2=1
Hence O is the mid point of CD.

  • 3

just find eq of both lines by 2 point form

then use section formula

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