Question

Find the ratio in which point P(k, 7) divides the segment joining A(8, 9) and B(1, 2). Also find k.

Answer 1

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 the ratio be x: 1.
Using the section formula,
$k=\frac{1x+1\times 8}{x+1}$
$\Rightarrow kx+k=x+8$................(1)
Also,$7=\frac{2x+9}{x+1}$
$\Rightarrow 7x+7=2x+9$
$\Rightarrow 5x=2$
$\Rightarrow x=\frac{2}{5}$
So, the required ratio is 2 : 5.
Putting this value of x in (1) we get
$k(\frac{2}{5}+1)=\frac{2}{5}+8$
$\Rightarrow \frac{7}{5}k=\frac{42}{5}$
$\Rightarrow k=6$