Question

The point dividing the line joining the points (1,2,3) and (3,-5,6) in the ratio 3:-5 is
A. $\left( {2,\dfrac{{ - 25}}{2},\dfrac{3}{2}} \right)$
B. $\left( { - 2,\dfrac{{25}}{2},\dfrac{{ - 3}}{2}} \right)$
C. $\left( {2,\dfrac{{25}}{2},\dfrac{3}{2}} \right)$
D. None of these

Answer 1

Hint: Use the section formula which will help us in finding the coordinates of a point dividing the line connecting $P({x_1},{y_1},{z_1})\,{\text{and }}Q({x_2},{y_2},{z_2})$ in the ratio $m:n$. The formula is $\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)$

Formula used: Section formula: $\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)$

Complete step by step solution:
We know from the section formula that the following are the coordinates of a point dividing the line connecting $P({x_1},{y_1},{z_1})\,{\text{and }}Q({x_2},{y_2},{z_2})$ in the ratio $m:n$-
$\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)$
Here,
${x_1} = 1,\,\,{y_1} = 2,\,\,{z_1} = 3,\,\,{x_2} = 3,\,\,{y_2} = - 5,\,\,{z_2} = 6,\,\,m = 3,\,\,n = - 5$
Substituting these values, we get the coordinates as
$\left( {\dfrac{{3(3) - 5(1)}}{{3 - 5}},\dfrac{{3( - 5) - 5(2)}}{{3 - 5}},\dfrac{{3(6) - 5(3)}}{{3 - 5}}} \right)$
$ = \left( {\dfrac{{9 - 5}}{{ - 2}},\dfrac{{ - 15 - 10}}{{ - 2}},\dfrac{{18 - 15}}{{ - 2}}} \right)$
$ = \left( { - 2,\dfrac{{25}}{2},\dfrac{{ - 3}}{2}} \right)$

Therefore, the correct option is option B. $\left( { - 2,\dfrac{{25}}{2},\dfrac{{ - 3}}{2}} \right)$

Note: There are two formulas for section formula, one where the point divides the line internally and the other where it divides externally. The coordinates of a point dividing the line connecting $P({x_1},{y_1},{z_1})\,{\text{and }}Q({x_2},{y_2},{z_2})$ in the ratio $m:n$ - internally is $\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)$ and externally is $\left(\dfrac{{m{x_2} - n{x_1}}}{{m - n}},\dfrac{{m{y_2} - n{y_1}}}{{m - n}},\dfrac{{m{z_2} - n{z_1}}}{{m - n}}\right)$.