Question

Find the coordinates of the point which divides internally the join of the points
(a) \[(8,9)\]and \[( - 7,4)\] in the ratio \[2:3\]
(b) \[(1, - 2)\]and \[(4,7)\] in the ratio \[1:2\].

Answer 1

Hint: In this question, since the line is divided internally in a ratio and the endpoints of the line segment are given, we will use the internal division method to find out the coordinates which divide the line segment. Also, as in this question, two different cases are given, so we have to apply the formula and solve both the cases separately.

Formula Used:
If a point$(x,y)$divides the line joining the points$({x_1},{y_1})$and$({x_2},{y_2})$in the ratio$m:n,$then
$(x,y) = \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)$

Complete Step-by-step solution
(a) Let us assume the point as $P(x,y)$
As per the points given in the question for (a),
${x_1} = 8,{y_1} = 9,{x_2} = - 7,{y_2} = 4,m = 2,n = 3$
Now, on substituting all the values in the formula mentioned above, we get:
$(x,y) = \left( {\dfrac{{(2 \times ( - 7)) + (3 \times 8)}}{{2 + 3}},\dfrac{{(2 \times 4) + (3 \times 9)}}{{2 + 3}}} \right)$
$(x,y) = \left( {\dfrac{{( - 14) + 24}}{5},\dfrac{{8 + 27}}{5}} \right)$
$(x,y) = \left( {\dfrac{{10}}{5},\dfrac{{35}}{5}} \right)$
$(x,y) = \left( {2,7} \right)$
Here, $x$ is 2 and $y$ is 7.
So, the point is $\left( {2,7} \right)$.

(b) Let us assume the point as $Q(x,y)$
As per the points given in the question for (b),
${x_1} = 1,{y_1} = - 2,{x_2} = 4,{y_2} = 7,m = 1,n = 2$
Now, on substituting all the values in the formula mentioned above, we get:
$(x,y) = \left( {\dfrac{{(1 \times 4) + (2 \times 1)}}{{1 + 2}},\dfrac{{(1 \times 7) + (2 \times ( - 2))}}{{1 + 2}}} \right)$
$(x,y) = \left( {\dfrac{{4 + 2}}{3},\dfrac{{7 + ( - 4)}}{3}} \right)$
$(x,y) = \left( {\dfrac{6}{3},\dfrac{{7 - 4}}{3}} \right)$
$(x,y) = \left( {\dfrac{6}{3},\dfrac{3}{3}} \right)$
$(x,y) = \left( {2,1} \right)$
Here, $x$ is 2 and $y$ is 1.
So, the point is $\left( {2,1} \right)$.
Hence the answer for (a) part is$\left( {2,7} \right)$and for (b) part is$\left( {2,1} \right)$.

Note:This type of question comes under the category of trisection of the line segment where a line segment can be divided internally and externally and the formula for both of them is different. If a line segment is divided internally, that means the point will come inside the line segment, whereas, if a line segment is divided externally, the line is extended so the line segment extends.