Question

In what ratio is a line segment joining the points (-2,-3) and (3,7) divided by $y$-axis?
${\text{A}}{\text{. }}\dfrac{2}{3}{\text{ }} \\\ {\text{B}}{\text{. }}\dfrac{3}{2}{\text{ }} \\\ {\text{C}}{\text{. }}\dfrac{4}{5}{\text{ }} \\\ {\text{D}}{\text{. }}\dfrac{5}{4}{\text{ }} \\\$

Answer 1

In this problem first we let the ratio in which $y$-axis divides the given line segment be in $m:n$.Then we apply the Section Formula on coordinates of that point to get relation between $m{\text{ and }}n$.And finally we put $x -$coordinate of point obtained from Section Formula to get ratio of $m{\text{ and }}n$.

Complete step by step answer:
Let $y$-axis divide the line segment joining the point (-2,-3) and (3,7) in the $m:n$ratio internally.
Since, the line segment is divided by $y$-axis so$x$ coordinate of the point is 0. Let the coordinates of the point which divides the line segment be ( 0,y)
Section Formula: If a point divides a line segment in $m:n$ whose endpoints coordinates are $({x_1},{y_1}){\text{ and }}({x_2},{y_2})$ then the coordinates of that point are
$\Rightarrow \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)$
Using Section Formula the coordinates of the point of division of line segment whose endpoints
(-2,-3) and (3,7) in $m:n$ratio.

\Rightarrow \left( {\dfrac{{m(3) + n( - 2)}}{{m + n}},\dfrac{{m(7) + n( - 3)}}{{m + n}}} \right) \\\ \Rightarrow \left( {\dfrac{{3m - 2n}}{{m + n}},\dfrac{{7m - 3n}}{{m + n}}} \right){\text{ }} \\\
eq.1
Since we let the coordinates of division points be $(0,y)$.
Then,
$\Rightarrow \left( {\dfrac{{3m - 2n}}{{m + n}},\dfrac{{7m - 3n}}{{m + n}}} \right) = (0,y)$
On comparing x coordinate of above equation we get
$\Rightarrow \dfrac{{3m - 2n}}{{m + n}} = 0 \\\ \Rightarrow 3m - 2n = 0 \\\ \Rightarrow 3m = 2n \\\ \Rightarrow \dfrac{m}{n} = \dfrac{2}{3} \\\$
Therefore, the ratio in which y axis divides the given line segment is $2:3$.
Hence, option A. is correct

Note:
Whenever you get this type of problem the key concept of solving is to have knowledge about
how a given point divides the line segment like in this question dividing point lies on y axis so its coordinates must be in form ( 0 ,y) . And using Section Formula $\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)$ you can find the required result.