Question

Coordinates of $P$ and $Q$ are $(4,-3)$ and $(-1,7)$. The abscissa of a point $R$ on the line segment $PQ$ such that $\frac{PR}{PQ}=\frac{3}{5}$ is

• A. $\frac{18}{5}$
• B. $\frac{17}{5}$
• C. $1$
• D. $\frac{17}{8}$

Answer 1

The correct option is C $1$

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

$\Rightarrow \frac{PQ}{PR}=\frac{5}{3}$......(1)

Here $PQ=PR+QR$

Substituting the value of $PQ$ in (1) and subtracting 1 from both sides

$\frac{PR+QR}{PR}-1=\frac{5}{3}-1$

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

So the ratio at which $R$ divides $PQ$ is $3:2$

We know that if a point $R(x,y)$ divides the line segment $PQ$ joining the points $P(x_1,y_1)$ and $Q(x_2,y_2)$ in the ratio $m:n$ then by the section formula.

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

As we are given the coordinates of $P$ and $Q$ and also $\frac{QR}{PR}$,

$x=\frac{3\times (-1)+2\times 4}{3+2},y=\frac{3\times 7+2\times (-3)}{3+2}$

$\therefore x=1,y=3$

So the abscissa of point R is 1