Question

the co ordinate of the point which divides internally the join of p (2,-1,4) and (4,3,2) in the ratio 2:3 is​

Answer 1

AnswEr:

Let R (x,y,z) be the required point.

(i) Let R divides PQ internally in the ratio 2:3, Then

$\\ \qquad \sf \: x = \frac{2 \times 4 + 3 \times 2}{2 + 3} \\ \\ \\ \qquad \sf \: y = \frac{2 \times 3 + 3 \times - 1}{2 + 3} \\ \\ \\ \qquad \sf \: z = \frac{2 \times 2 + 3 \times 4}{2 + 3} \\ \\ \\ \implies \sf \: x = \frac{14}{5} \: , \: y = \frac{3}{5} \: , \: z = \frac{16}{5}$

So, the coordinates of the points are

$\\ \qquad \sf \: ( \frac{14}{5} , \frac{3}{5} , \frac{16}{5} )$

Answer 2

$\huge \fcolorbox{red}{pink}{Solution :)}$

Let ,

The coordinate of the point which divides internally the join of p (2,-1,4) and (4,3,2) be R(x,y,z)

We know that , the coordinates of point R which divides the line segment joining the two points P(x1,x2,x3) and Q(x1,x2,x3) internally in the ratio m : n is given by

$\large{ \star} \: \: \sf \large \fbox{ \frac{mx_{2} + nx_{1}}{m + n} , \frac{my_{2} + ny_{1}}{m + n} , \frac{mz_{2} + nz_{1}}{m + n} }$

Substitute the known values , we get

$\sf \mapsto (x, y,z) = \frac{2(4) + 3(2)}{2 + 3} , \frac{2(3) + 3( - 1)}{2 + 3} , \frac{2(2) + 3(4)}{2 + 3} \\ \\ \sf \mapsto (x, y,z) = \frac{8 + 6}{5} , \frac{6 - 3}{5} , \frac{4 + 12}{5} \\ \\ \sf \mapsto (x, y,z) = \frac{14}{5} , \frac{3}{5} , \frac{16}{5}$

Hence , the required coordinate of R is (14/5 , 3/5 , 16/5)

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