Question

In which ratio Y-axis divides the segment joining the points \[A\left( {5, - 6} \right)\] and \[B\left( { - 1, - 4} \right)\].

Answer 1

Hint:
To find the ratio in which a point divides the line segment given between two points, we use the section formula. Here it is given that the Y-axis divides, hence the abscissa(x-coordinate) of the point dividing the line is 0, Hence, we will use section formula for the abscissa to find the ratio and get to the final answer.

Complete step by step solution:
We know that according to section formula if a point $(x,y)$ whose endpoints are $({x_1},{y_1})$ and $({x_2},{y_2})$ divides a line segment in ratio m:n, then the point which divide has
The x-coordinate (abscissa) as
$x = \dfrac{{{x_1}n + {x_2}m}}{{m + n}}$ … (1)
The y-coordinate (ordinate) as
$y = \dfrac{{{y_1}n + {y_2}m}}{{m + n}}$ … (2)
We know that on the y-axis, the abscissa (x- coordinate) is always 0.
Since we are given that the line is divided by the y-axis Hence, we know that abscissa of this given point must be 0, Hence we will use (1)
Now, using (1), we get
$ \Rightarrow x = \dfrac{{{x_1}n + {x_2}m}}{{m + n}}$
We are given that the endpoints of line segments are \[A\left( {5, - 6} \right)\] and \[B\left( { - 1, - 4} \right)\]
Hence,
\[{x_1} = 5\] and \[{x_2} = - 1\]
On substituting \[x = 0\] , \[{x_1} = 5\] and \[{x_2} = - 1\] , we get
$ \Rightarrow 0 = \dfrac{{5 \times n + ( - 1) \times m}}{{m + n}}$
On cross-multiplying, we get
$ \Rightarrow 0 \times (m + n) = 5n + ( - 1)m$
Since any number multiplied by 0 results in 0, we get
$ \Rightarrow 0 = 5n - m$
On shifting the term containing m to LHS, we get
$ \Rightarrow m = 5n$
On rearranging the terms to get the ratio as $\dfrac{m}{n}$ we get
$ \Rightarrow \dfrac{m}{n} = \dfrac{5}{1}$

Hence, we get the required ratio $\dfrac{m}{n} = \dfrac{5}{1}$.

Note:
In these questions, try to find either the abscissa or the ordinate of the point which divides the, we can solve the whole question even if we know any one of it. Though in this case we luckily got the point to internally divide the line, this formula can also be used when the point externally divides the line, the only difference would be that the calculated ratio would be negative.