Question

Find the ratio in which the point (−3, k) divides the line-segment joining the points (−5, −4) and (−2, 3). Also find the value of k ?

Answer 1

Suppose the point P(−3, k) divides the line segment joining the points A(−5, −4) and B(−2, 3) in the ratio m : 1.

Then, the coordinates of the point P will be 

\[\left( \frac{- 2m - 5}{m + 1}, \frac{3m - 4}{m + 1} \right)\]

Also, it is given that the coordinates of the point P are (−3, k).

\[\therefore \frac{- 2m - 5}{m + 1} = - 3\ \text{and}\ \frac{3m - 4}{m + 1} = k\]
\[ \Rightarrow - 2m - 5 = - 3m - 3\ \text{and}\ \frac{3m - 4}{m + 1} = k\]
\[ \Rightarrow m = 2\ \text{and}\ k = \frac{3m - 4}{m + 1}\]
\[ \Rightarrow m = 2\ \text{and}\ k = \frac{2}{3}\]

Hence, the required ratio is 2 : 1 and the value of k is

\[\frac{2}{3}\]