# Division of a straight line in given ratio

The problem in the book states:

"Find the coordinates of the point that divides AB in the given ratio in each of the following cases: a) A(2,4), B(-3,9) 1:4 internally"

Given the point that we are looking for is $P(X,Y)$.

Since I am not familiar with the "line division" conventions, I went to find a point (-2,8) with the following formula: $$X=\frac{\lambda x_2+\mu x_1}{\lambda+\mu}$$, where $\lambda =1$ and $\mu=4$. And a similar formula is used to find coordinate of $Y$.

I was having a cartesian coordinate system in mind when I was dividing the line into ratios. So that the portion on the left is 1 unit and the portion on the right is 4 units. However, the book assumed inversely, so that:

Would you say that my answer is incorrect, or both answers are correct? I would imagine that it would be more accurate for the book to give the ratio 5:-1, if it wanted me to arrive at the answer they are giving.

• Probably the ratio 1:4 is to be taken not from left to right, but from the first point given (A) to the second one (B). However your answer is correct, because the book should have specified more clearly what was intended. Sep 13 '15 at 14:23
• MyPoint=$P_1$ is such that $BP_1/AP_1=1/4$. TheirPoint $P_2$ is such that $AP_2/BP_2 =1/4$. So, without specifing the order of the ratio we have two solutions. Sep 13 '15 at 14:32

Notice, the coordinates of the point $P(X, Y)$ which divides the line joining the points $A(2, 4)$ & $B(-3, 9)$ in a ratio $1:4$ internally are given as $$P(X, Y)\equiv \left(\frac{4(2)+1(-3)}{1+4}, \frac{4(4)+1(9)}{1+4}\right)$$ $$P(X, Y)\equiv \color{blue}{\left(1, 5\right)}$$
$\dfrac{BP}{PA}= \dfrac14$ your point,
$\dfrac{BQ}{QA}=\dfrac41$ their point... is correct as per the convention.