If the point P(2,1) lies on the line segment joining points A(4, 2) and 6(8, 4), then
(a) $A P=\frac{1}{3} A B$
(b) $A P=P B$
(c) $P B=\frac{1}{3} A B$
(d) $A P=\frac{1}{2} A B$
(d) Given that, the point P(2,1) lies on the line segment joining the points A(4,2) and 8(8, 4), which shows in the figure below:
Now, distance between $A(4,2)$ and $(2,1), A P=\sqrt{(2-4)^{2}+(1-2)^{2}}$
$=\sqrt{(-2)^{2}+(-1)^{2}}=\sqrt{4+1}=\sqrt{5}$
Distance between $A(4,2)$ and $B(8,4)$,
$A B=\sqrt{(8-4)^{2}+(4-2)^{2}}$
$=\sqrt{(4)^{2}+(2)^{2}}=\sqrt{16+4}=\sqrt{20}=2 \sqrt{5}$
Distance between $B(8,4)$ and $P(2,1), B P=\sqrt{(8-2)^{2}+(4-1)^{2}}$
$=\sqrt{6^{2}+3^{2}}=\sqrt{36+9}=\sqrt{45}=3 \sqrt{5}$
$\therefore$ $A B=2 \sqrt{5}=2 A P \Rightarrow A P=\frac{A B}{2}$
Hence, required condition is $A P=\frac{A B}{2}$