Question

Find the coordinates of the point $R$ on the line segment joining the points $P(1,3)$ and $Q(2,5)$ such that $PR=\frac{3}{5}PQ$.

• A. $\frac{8}{5},\frac{21}{5}$
• B. $\frac{8}{5},\frac{19}{5}$
• C. $\frac{7}{5},\frac{19}{5}$
• D. $\frac{7}{5},\frac{21}{5}$

Answer 1

The correct option is A $\frac{8}{5},\frac{21}{5}$

Given that the point $R(x,y)$ divides the line segment $PQ$ joining the points $P(x_1,y_1)=(1,3)\&Q(x_2,y_2)=(2,5)$ in a ratio such that $\frac{PR}{PQ}=\frac{3}{5}$ i.e $PR<PQ$.

$\Rightarrow R$ divides $PQ$ internally.

Here $\frac{PR}{PQ}=\frac{3}{5}$

$\Rightarrow \frac{PQ}{PR}=\frac{5}{3}$

$\Rightarrow \frac{PR+RQ}{PR}=\frac{5}{3}$

$\Rightarrow \frac{RQ}{PR}=\frac{2}{3}$

$\Rightarrow \frac{PR}{QR}=\frac{3}{2}$

$\therefore m:n=3:2$

$\therefore$ By the section formula, we have

$x=\frac{n{x}_1+m{x}_2}{m+n}\&y=\frac{n{y}_1+m{y}_2}{m+n}$

i.e $x=\frac{2\times 1+3\times 2}{2+3}=\frac{8}{5}$

and $y=\frac{2\times 3+3\times 5}{2+3}=\frac{21}{5}$

Therefore, $x=\frac{8}{5};y=\frac{21}{5}$

Hence, option A is the correct answer.