Question

In what ratio does the point (-4,6) divide the line segment joining the points $A\left( { - 6,10} \right)$ and $B\left( {3, - 8} \right)$.

Answer 1

Hint:
Here, we are required to find in what ratio does the given point divide the given line segment. We will use the Section formula, to find the required ratio. Since, we are required to find the ratio, we will let the ratio be $k:1$ as this makes the question easy to solve. Substituting the values in the section formula, we will get the required ratio.

Formula Used:
Section Formula:
Coordinates of point $C = \dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}$, where $C$ is a point dividing the line segment.

Complete step by step solution:
According to the question,
We are given a line segment $AB$ such that the coordinates of Point $A = \left( { - 6,10} \right)$ and the coordinates of point $B = \left( {3, - 8} \right)$.
Now, there is a point $C$ having the coordinates $\left( { - 4,6} \right)$ which divides the line segment $AB$.
Let us assume that point $C$ divides the line segment $AB$in the ratio $k:1$, where $k$ is a constant.

Now, by section formula, we know that,
Coordinates of Point $C = \dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}$………………………………(1)
Where, $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ are the coordinates of the end points of the line segment respectively.
Also, $m$and $n$ are the ratios in which the points have divided the line segment.
Now, substituting $\left( {{x_1},{y_1}} \right) = \left( { - 6,10} \right)$ , $\left( {{x_2},{y_2}} \right) = \left( {3, - 8} \right)$ and the ratio $m:n = k:1$ in equation (1), we get,
Coordinates of point $C = \dfrac{{k\left( 3 \right) + 1\left( { - 6} \right)}}{{k + 1}},\dfrac{{k\left( { - 8} \right) + 1\left( {10} \right)}}{{k + 1}}$
But coordinates of point $C = \left( { - 4,6} \right)$
$ \Rightarrow \left( { - 4,6} \right) = \dfrac{{3k - 6}}{{k + 1}},\dfrac{{10 - 8k}}{{k + 1}}$
Now, let us compare the $x$ coordinates,
$ \Rightarrow - 4 = \dfrac{{3k - 6}}{{k + 1}}$
Taking denominator to the LHS, we get,
$ \Rightarrow - 4k - 4 = 3k - 6$
Solving further,
$ \Rightarrow 7k = 2$
Dividing both sides by 7,
$ \Rightarrow k = \dfrac{2}{7}$
Hence, the ratio $k:1$ becomes $\dfrac{2}{7}:1$
Or
$k:1 = 2:7$
Therefore, the point $C\left( { - 4,6} \right)$ divides the line segment joining the points $A\left( { - 6,10} \right)$ and $B\left( {3, - 8} \right)$in the ratio $2:7$ internally.

Hence, this is the required answer.

Note:
In this question, three points are given with their respective coordinates. We should take care while solving the question, that we substitute the correct coordinates in the correct place. For example, in the section formula, if we substitute $\left( {{x_1},{y_1}} \right)$ in such a way that the $x$ coordinate is of point $A$ and the $y$coordinate is of point $B$. Then, our answer will be wrong. Also, after finding the ratio, we should know that $2:7$ is different from $7:2$, hence, we should find the ratio accordingly.