Question 18
Find the coordinates of the point R on the line segment joining the points P(-1,3) and Q(2,5) such that
$P R = \frac{3}{5} P Q .$

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Solution

According to the question,

Given: $P R = \frac{3}{5} P Q$
$\Rightarrow \frac{P Q}{P R} = \frac{5}{3} \Rightarrow \frac{P R + R Q}{P R} = \frac{5}{3} \Rightarrow 1 + \frac{R Q}{P R} = \frac{5}{3} \Rightarrow \frac{R Q}{P R} = \frac{5}{3} - 1 = \frac{2}{3} ∴ R Q : P R = 2 : 3 O r P R : R Q = 3 : 2$

Suppose, R(x,y) be the point which divides the line segment joining the points P(-1,3) and Q(2,5) in the ratio 3:2.

$∴ (x, y) = {\frac{3 (2) + 2 (- 1)}{3 + 2}, \frac{3 (5) + 2 (3)}{3 + 2}}$

$[\begin{matrix} ∵ B y i n t e r n a l s e c t i o n f o r m u l a, {\frac{m_{2} x_{1} + m_{1} x_{2}}{m_{1} + m_{2}}, \frac{m_{2} y_{1} + m_{1} y_{2}}{m_{1} + m_{2}}} \end{matrix}]$

$= (\frac{6 - 2}{5}, \frac{15 + 6}{5}) = (\frac{4}{5}, \frac{21}{5})$ Hence, the required coordinates of the point R is $(\frac{4}{5}, \frac{21}{5})$

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