In what ratio is the line segment joining $(-3, -1)$ and $(-8, -9)$ divided at the point $(-5, −\frac{21}{5})$?
Given:
The line segment joining the points $(-3, -1)$ and $(-8, -9)$ is divided at the point $(-5, −\frac{21}{5}).
To do:
We have to find the ratio of the division.
Solution:
Let the point \( \left(-5, \frac{-21}{5}\right) \) divides the line segment joining the points \( (-3,-1) \) and \( (-8,-9) \) in the ratio of \( m: n \).
Using section formula, we have,
\( (x, y)=(\frac{mx_{2}+nx_{1}}{m+n}, \frac{my_{2}+ny_{1}}{m+n}) \)
Therefore,
\( x=\frac{mx_{2}+nx_{1}}{m+n} \)
\( \Rightarrow -5=\frac{m(-8)+n(-3)}{m+n} \)
\( \Rightarrow -5=\frac{-8m-3n}{m+n} \)
\( \Rightarrow -5m-5n=-8m-3n \)
\( \Rightarrow -5 m+8 m=-3 n+5 n \)
\( \Rightarrow 3 m=2 n \)
\( \Rightarrow \frac{m}{n}=\frac{2}{3} \)
\( \Rightarrow m:n=2:3 \)
The required ratio is $2:3$.
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