Find the ratio in which the y-axis divides line segment joining the points $(  -4,\ -6)$ and $( 10,\ 12)$. Also find the coordinates of the point of division.

Given: A line segment joining the points $( -4,\ -6) \ and\ ( 10,\ 12)$.

To do: To find the ratio in which y-axis divides the line and to find the coordinates of the point off division.

Solution:
 Let the y-axis divide the line segment joining the points $( -4,\ 6)$ and $( 10,\ 12)$ in the ratio k: 1 and the point of the intersection be $( 0,\ y)$. 

Using section formula, we have$P( x,\ y) =( \frac{nx_{1} +mx_{2}}{m+n} ,\ \frac{ny_{1} +my_{2}}{m+n})$

$( 0,\ y) =(\frac{1\times -4+k\times 10}{k+1} ,\ \frac{-6\times 1+k\times 12}{k+1})$

$\Rightarrow \frac{10k-4}{k+1} =0\ \ \ \ .......( 1) \ and\ \frac{12k-6}{k+1} =y\ \ \ \ .........( 2)$

$\Rightarrow 10k-4=0$

$\Rightarrow 10k=4$

$\Rightarrow k=\frac{4}{10} =\frac{2}{5}$ on\ subtituting this value in $( 2)$

$\Rightarrow y=\frac{12\times \frac{2}{5} -6}{\frac{2}{5} +1}$

$=\frac{-6}{7}$

Therefore y-axis divides the line in the ratio of 2:5 and the point of intersection is $( 0,\ -\frac{6}{7})$.

 

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Updated on: 10-Oct-2022

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