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Point $P( x,\ 4)$ lies on the line segment joining the points $A( -5,\ 8)$ and $B( 4,\ -10)$. Find the ratio in which point P divides the line segment AB. Also find the value of x.
Given: A line segment AB joining the points $A( -5,\ 8)$ and $B( 4,\ -10)$ and a point $P( x,\ 4)$
To do: To find the ratio in which P divides the given line segment AB and to find the value of x.
Solution: Let us say that point $P( x,\ 4)$ divides the given line segment in m:n.
we know that if there is a line segment joining two points$\ A( x_{1} ,\ y_{1}) \ $and $B( x_{2} ,\ y_{2})$ and if there is a point$\ P( x,\ y)$ lies on the line segment and divides the line segment in a ratio of m:n, then
we have $P( x,\ y) =\left(\frac{nx_{1} +mx_{2}}{m+n} ,\ \frac{ny_{1} +my_{2}}{m+n}\right)$
$\therefore P( x,4) =\left(\frac{4m-5n}{m+n} ,\ \frac{-10m+8n}{m+n}\right)$
$\Rightarrow x=\frac{4m-5n}{m+n} \ \ and\ 4=\frac{-10m+8n}{m+n}$
$\Rightarrow x=\frac{4m-5n}{m+n} \ \ and\ 4=\frac{-10m+8n}{m+n}$
$\Rightarrow 4m+4n=-10m+8n$
$\Rightarrow 4m+10m=8n-4n$
$\Rightarrow 14m=4n$
$\Rightarrow \frac{m}{n} =\frac{4}{14} =\frac{2}{7}$
$m:n=2:7$
And $x=\frac{4m-5n}{m+n}$
$x=\frac{4\left(\frac{m}{n}\right) -5\left(\frac{n}{n}\right)}{\frac{m}{n}+\frac{n}{n}}$ $( on\ divididing\ both\ numerator\ and\ denominator\ by\ n)$
$=\frac{4\left(\frac{2}{7}\right) -5( 1)}{\frac{2}{7} +1}$
$=\frac{\frac{8-35}{7}}{\frac{2+7}{7}}$
$=\frac{-27}{9}$
$=-3$
Thus point P divides the line in the ratio of $2:7$ and the value of $x$ is $-3$.
Updated on: 10-Oct-2022
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