Question

Find the ratio in which the line segment joining the points $( - 3,10)$ and $(6, - 8)$ is divided by $( - 1,6)$

Answer 1

Hint:For this question we have to know the section formula then it will be easy to solve. The point ${\text{P}}$ divides the line segment ${\text{AB}}$ into two parts ${\text{AP}}$ and ${\text{PB}}$. In other words, the midpoint ${\text{P}}$ divides the line segment ${\text{AB}}$ . We need to find the ratio between ${\text{AP}}$ and ${\text{PB}}$.

Formula used:The section formula used for this question is,
The co-ordinates of \[{\text{P(x,y) = }}\left( {\left. {\dfrac{{{\text{m}}{{\text{x}}_2} + {\text{n}}{{\text{x}}_1}}}{{{\text{m}} + {\text{n}}}},\dfrac{{{\text{m}}{{\text{y}}_2} + {\text{n}}{{\text{y}}_1}}}{{{\text{m}} + {\text{n}}}}} \right)} \right.\]
Where,
${\text{(}}{{\text{x}}_1},{{\text{y}}_1}{\text{)}}$ be the co-ordinates of point ${\text{A}}$
$({{\text{x}}_2}{\text{,}}{{\text{y}}_2}{\text{)}}$ be the co-ordinates of point ${\text{B}}$
${\text{m}}$ and ${\text{n}}$ be the ratio of line segment joining the points

Complete step-by-step answer:
The data given in the question,

Let assume that the point ${\text{P(}} - 1,6{\text{)}}$ joining the two points ${\text{A(}} - 3,10{\text{)}}$ and ${\text{B(}}6, - 8{\text{)}}$ are in the ratio of \[{\text{k:}}1\]
Then
\[{{\text{x}}_1} = - 3\] and ${{\text{y}}_1} = 10$
${{\text{x}}_2} = 6$ and ${{\text{y}}_2} = - 8$
${\text{x}} = - 1$ and ${\text{y}} = 6$
${\text{m}} = {\text{k}}$ and ${\text{n}} = 1$
Using the section formula,
Substituting all the values in the coordinates of point ${\text{P}}$,
\[{\text{P}}\left[ {\left. {\dfrac{{{\text{k(}}6{\text{)}} + 1{\text{(}} - 3{\text{)}}}}{{{\text{k}} + {\text{1}}}},\dfrac{{{\text{k}}( - 8){\text{ + }}1{\text{(}}10{\text{)}}}}{{{\text{k}} + {\text{1}}}}} \right]} \right.\]
While solving the above co-ordinates we get,
$ \Rightarrow {\text{P}}\left[ {\left. {\dfrac{{6{\text{k}} - 3}}{{{\text{k}} + {\text{1}}}},\dfrac{{ - 8{\text{k}} + 10}}{{{\text{k}} + {\text{1}}}}} \right]} \right.$
From the given data the coordinates of point are ${\text{P(}} - 1,6{\text{)}}$
While taking the ${\text{x}}$ coordinate of point ${\text{P}}$,
$ \Rightarrow - 1 = \dfrac{{6{\text{k}} - 3}}{{{\text{k}} + 1}}$
By doing cross multiplication we get,
$ \Rightarrow - 1{\text{(k}} + 1{\text{)}} = 6{\text{k}} - 3$
Making the ${\text{k}}$ term one side and constant term on other side we get,
$ \Rightarrow 6{\text{k}} + 1{\text{k}} = - 1 + 3$
While solving the above equation we get,
$ \Rightarrow {\text{7k}} = 2$
Then the value of ${\text{k}}$is,
$ \Rightarrow {\text{k}} = \dfrac{2}{7}$
$\therefore $ The required ratio is $2:7$
Hence, the ratio in which the line segment joining the points $( - 3,10)$ and $(6, - 8)$ is divided by $( - 1,6)$ is $2:7$.

Note:We have taken ${\text{x}}$ coordinate of point ${\text{P(}} - 1,6{\text{)}}$ to solve this question. We can also take ${\text{y}}$ coordinate of point ${\text{P(}} - 1,6{\text{)}}$ to solve this and the same ratio will be the result. By using the same section formula, if the ratio is given, we can also find the coordinates of point ${\text{P}}$.