Question

If $P$ is a point lying on the line segment joining $Q(1,1)$ and $R(4,1)$ such that $PQ=\frac{3}{7}QR$, then find the coordinates of $P$.

Answer 1

Compute the required coordinates:

Given,

$Q=(1,1)$

$R=(4,1)$

$P$ lies on $QR$.

$PQ=\frac{3}{7}QR$

So, $PR=1-\frac{3}{7}$

$\Rightarrow$ $PR=\frac{4}{7}QR$

$\Rightarrow$$PQ:PR=3:4$

Therefore, $P$ divides $QR$ in ratio of $3:4$.

Using 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;

$\left(x,y\right)=\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right)$

$P=$$\left(\frac{3\times 4+4\times 1}{3+4},\frac{3\times 1+4\times 1}{3+4}\right)$

$=$$\left(\frac{16}{7},1\right)$

Hence the coordinates of $P$ is $\left(\frac{16}{7},1\right)$.