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  • Find the coordinates of the point R on the line segment joining the points P(–1, 3) and Q(2, 5) such that PR = 3 by 5 PQ.

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

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Solution
                 
section\, formula \left ( \frac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}},\frac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}} \right )
According to question let R = (x, y) and PR = \frac{3}{5} PQ
\frac{PQ}{PR}= \frac{3}{5}
R lies on PQ                \therefore PQ = PR +RQ

\frac{PR+PQ}{PR}= \frac{5}{3}
On dividing separately we get
1+\frac{RQ}{PR}= \frac{5}{3}
\frac{RQ}{PR}= \frac{5}{3}-1= \frac{2}{3}
\Rightarrow PR:RQ= 3:2
Hence, R divides PQ in ratio 3 : 2 using section formula we have
(x1, y1) = (-1, 3)                       (x2, y2) = (2, 5)
m1 = 3, m2 = 2
R\left ( x,y \right )= \left ( \frac{3\left ( 2 \right )+2\left ( -1 \right )}{3+2} ,\frac{3\left ( 5 \right )+2\left ( 3 \right )}{3+2}\right )
R\left ( x,y \right )= \left ( \frac{6-2}{5},\frac{15+6}{5} \right )
R\left ( x,y \right )= \left ( \frac{4}{5},\frac{21}{5} \right )
Here co- ordinates of R is \left ( \frac{4}{5},\frac{21}{5} \right )
           

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