If the point P(2,1) lies on the line segment joining points A(4,2) and B(8,4) then
AP=12AB
Given that, the point P(2,1) lies on the line segment joining points A(4,2) and B(8,4) which shows in the figure below:
Now, distance between
A(4,2) and P(2,1), AP=√(2−4)2+(1−2)2
[∵ Distance between two points (x1,y1) and (x2,y2),d=√(x2−x1)2+(x2−y1)2]
=√(−2)2+(−1)2=√4+1=√5
Distance between A(4,2) and B(8,4),
AB=√(8−4)2+(4−2)2
=√(4)2+(2)2=√16+4=√20=2√5
Distance between B(8,4) and P(2,1), BP
=√(8−2)2+(4−1)2
=√62+32=√36+9=√45=3√5
∴AB=2√5=2AP⇒AP=AB2
Hence, required condition is AP =AB2