Question

Find the ratio in which the point \[\left( -3,k \right)\] divides the line segment joining the points \[\left( -5,-4 \right)\] and \[\left( -2,3 \right)\]. Also find the value of k.

Answer 1

Hint: In this question, we first need to assume the ratio in which the given point divides the line segment formed by the other two points as \[m:1\]. Now, using the section formula given by \[\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)\] on equating the x coordinate on both the sides we get the ratio. Then on substituting the values obtained in the y coordinate and equating we get the value of k.

Complete step-by-step solution -
SECTION FORMULA:
The coordinate of the point which divides the joint of \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] in the ratio \[m:n\] internally is
\[\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)\]
Now, given in the question that \[\left( -3,k \right)\] divides the joint of \[\left( -5,-4 \right)\] and \[\left( -2,3 \right)\]
Now, on comparing the above points with the section formula we get,
\[{{x}_{1}}=-5,{{x}_{2}}=-2,{{y}_{1}}=-4,{{y}_{2}}=3\]
Now, let us assume that the point \[\left( -3,k \right)\] divides the line segment in the ratio \[m:1\]
Now, from the section formula we have,
\[\Rightarrow \left( -3,k \right)=\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)\]

Let us now substitute the respective values of the coordinates and the ratio in the above formula and simplify further
\[\Rightarrow \left( -3,k \right)=\left( \dfrac{m\left( -2 \right)-5}{m+1},\dfrac{m\left( 3 \right)-4}{m+1} \right)\]
Now, this can be further written in the simplified form as
\[\Rightarrow \left( -3,k \right)=\left( \dfrac{-2m-5}{m+1},\dfrac{3m-4}{m+1} \right)\]
Let us now equate the x coordinates in both the points of the above equation
\[\Rightarrow -3=\dfrac{-2m-5}{m+1}\]
Now, on cross multiplication on both the sides we get,
\[\Rightarrow -3\left( m+1 \right)=-2m-5\]
Now, on multiplying the respective terms and simplifying further we get,
\[\Rightarrow -3m-3=-2m-5\]
Let us now rearrange the terms on both the sides
\[\Rightarrow 3m-2m=5-3\]
Now, on further simplification we get,
\[\therefore m=2\]
Thus, the point divides the line segment in the ratio \[2:1\]
Now, let us substitute the value of m in the section formula and equate the y coordinates
\[\Rightarrow k=\dfrac{3m-4}{m+1}\]
Now, on substituting the value of m in the above equation we get,
\[\Rightarrow k=\dfrac{3\left( 2 \right)-4}{2+1}\]
Now, this can be further written as
\[\Rightarrow k=\dfrac{6-4}{3}\]
Now, on further simplification we get,
\[\therefore k=\dfrac{2}{3}\]
Hence, the point \[\left( -3,\dfrac{2}{3} \right)\] divides the line segment in the ratio \[2:1\]

Note: Instead of assuming the ratio in which the point divides the line segment as \[m:1\] we can also solve it by assuming the ratio as \[m:n\] and simplify further which gives the value of \[\dfrac{m}{n}\] so that we get the values of m and n. Here, the value of m and n will be the same in both the methods. It is important to note that we considered the ratio as \[m:1\] instead of \[m:n\] because when we substitute we have three unknowns m, n and k but only two equations. So, we take \[m:1\] because even if there is a value for n then we will get it in the fraction of m. So, to avoid assumptions we can directly consider \[m:1\].