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Co-ordinates of the point P dividing the line segment joining the points A(1, 3) and B(4, 6), in the ratio 2:1 are:
$( A) \ ( 2,4)$
$( B) \ ( 3,\ 5)$
$( C) \ ( 4,\ 2)$
$( D) \ ( 5,\ 3)$
Given: A line segment AB, joining the points $A( 1,\ 3)$ and $B( 4,\ 6)$ and a point P lies on the given line segment AB dividing the line segment in the ratio 2:1.
To do: To find out the co-ordinates of the given line segment.
Solution: We know that if there is line segment AB joining two points $A( x_{1} ,y_{1})$ and $B( x_{2} ,y_{2})$ and there is a point $P( x,\ y)$ lying on the line segment dividing in the ratio m:n
Then Using section formula, we have, $P( x,\ y) =\left(\frac{nx_{1} +mx_{2}}{m+n} ,\ \frac{ny_{1} +my_{2}}{m+n}\right)$
Here we have, $\ x_{1} =1,x_{2} =4,y_{1} =3\ and\ y_{2} =6\ ,m=2\ and\ n=1$
$\therefore \ P( x,\ y) =\left(\frac{1\times 1+2\times 4}{2+1} ,\ \frac{1\times 3+2\times 6}{2+1}\right)$
$=\left(\frac{9}{3} ,\ \frac{15}{3}\right)$
$=( 3,5)$
$\therefore$ Option $( B)$ is correct.
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