Question

If the point P (2, 1) lies on the segment joining Points A (4, 2) and B (8, 4) then

• A. $AP=\frac{1}{3}AB$
• B. AB = PB
• C. $PB=\frac{1}{3}AB$
• D. $AP=\frac{1}{2}AB$

Answer 1

The correct option is D $AP=\frac{1}{2}AB$

Using the section formula, if a point $(x,y)$ divides the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$, then

$(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$
Let $P$ divide $AB$ in the ratio $k:1$

Substituting $(x_1,y_1)=(4,2)$ and $(x_2,y_2)=(8,4)$ in the section formula, we get

$P=(\frac{k\left(8\right)+1\left(4\right)}{k+1},\frac{k\left(4\right)+1\left(2\right)}{k+1})$

But given $P=(2,1)$

$\Rightarrow (\frac{8k+4}{k+1},\frac{4k+2}{k+1})=(2,1)$

Comparing the x - coordinate,
$\Rightarrow \frac{8k+4}{k+1}=2$
$\Rightarrow 8k+4=2k+2$

$6k=-2$

$k=-\frac{1}{3}$

As $k$ is negative, $P$ divides $AB$ in the ratio $1:3$ externally.

$\Rightarrow \frac{AP}{PB}=\frac{1}{3}$

$\Rightarrow AP=\frac{1}{2}AB$