Question
Find the ratio in which the point $P\left(\frac{3}{4}, \frac{5}{12}\right)$ divides the line segment joining the points $A\left(\frac{1}{2}, \frac{3}{2}\right)$ and $B (2,-5)$.
✓
Answer
Let $P$ divides the line segment joining the points $A$ and $B$ in the ratio $m: n$.
$A \left(x_1, y_1\right) P (x, y) B \left(x_2, y_2\right)$
Then the coordinates of point $P$ is given by the section formula,
$P=\left(\frac{m x_2+n x_1}{m+n}, \frac{m y_2+n y_1}{m+n}\right)$
Where $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ are the coordinates of the points joined to make the line segment.
Here $P=\left(\frac{3}{4}, \frac{5}{12}\right)$.
Let $\left(x_1, y_1\right)=\left(\frac{1}{2}, \frac{3}{2}\right)$ and $\left(x_2, y_2\right)=(2,-5)$
Using the section formula we get,
$\left(\frac{3}{4}, \frac{5}{12}\right)=\left(\frac{2 m+\frac{1}{2} n}{m+n}, \frac{-5 m+\frac{3}{2} n}{m+n}\right)$
Comparing the $x$ and $y$ coordinates of both the sides.
$\Rightarrow \frac{3}{4}=\frac{2 m+\frac{1}{2} n}{m+n}$ and $\frac{5}{12}=\frac{-5 m+\frac{3}{2} n}{m+n}$
Using $\frac{3}{4}=\frac{2 m+\frac{1}{2} n}{m+n}$ we get,
$3(m+n)=4\left(2 m+\frac{1}{2} n\right)$
$\Rightarrow 3 m+3 n=8 m+2 n$
$\Rightarrow 3 n-2 n=8 m-3 m$
$\Rightarrow n=5 m$
Hence, $P$ divides the given line segment in the ratio $1: 5.$
$A \left(x_1, y_1\right) P (x, y) B \left(x_2, y_2\right)$
Then the coordinates of point $P$ is given by the section formula,
$P=\left(\frac{m x_2+n x_1}{m+n}, \frac{m y_2+n y_1}{m+n}\right)$
Where $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ are the coordinates of the points joined to make the line segment.
Here $P=\left(\frac{3}{4}, \frac{5}{12}\right)$.
Let $\left(x_1, y_1\right)=\left(\frac{1}{2}, \frac{3}{2}\right)$ and $\left(x_2, y_2\right)=(2,-5)$
Using the section formula we get,
$\left(\frac{3}{4}, \frac{5}{12}\right)=\left(\frac{2 m+\frac{1}{2} n}{m+n}, \frac{-5 m+\frac{3}{2} n}{m+n}\right)$
Comparing the $x$ and $y$ coordinates of both the sides.
$\Rightarrow \frac{3}{4}=\frac{2 m+\frac{1}{2} n}{m+n}$ and $\frac{5}{12}=\frac{-5 m+\frac{3}{2} n}{m+n}$
Using $\frac{3}{4}=\frac{2 m+\frac{1}{2} n}{m+n}$ we get,
$3(m+n)=4\left(2 m+\frac{1}{2} n\right)$
$\Rightarrow 3 m+3 n=8 m+2 n$
$\Rightarrow 3 n-2 n=8 m-3 m$
$\Rightarrow n=5 m$
Hence, $P$ divides the given line segment in the ratio $1: 5.$
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