Question

 Find the ratio in which the line segment joining the points $A\left( {3, - 3} \right)$ and $B\left( { - 2,7} \right)$ is divided by the $x - $ axis. Also find the coordinates of the point of division.

Answer 1

Hint – In order to solve this question, we will use section formula i.e. $\left( {x = \dfrac{{{m_1}{x_2} + {m_2}{x_1}}}{{{m_1} + {m_2}}}{\text{ }},{\text{ }}y = \dfrac{{{m_1}{y_2} + {m_2}{y_1}}}{{{m_1} + {m_2}}}} \right)$ . In this way we will get our desired answer.

Complete Step-by-Step solution:
Now given points are,
$A\left( {3, - 3} \right)$ and $B\left( { - 2,7} \right)$.

 Now we have to find out the coordinates of the point of division by using the section formula.
Section formula-This formula tells us the coordinates of the point which divides a given line segment into two parts such that their lengths are in the ratio m: n.
$\left( {x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}}{\text{ }},{\text{ }}y = \dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)$ .
Here, $\left( {{x_1} = 3,{y_1} = - 3} \right)$ , $\left( {{x_2} = - 2,{y_2} = 7} \right)$ .
And m:n$ = $k:1
Given that the points are divided by $x - $ axis i.e. $P\left( {x,0} \right)$ and we know that the $y$ coordinate of $x - $ axis is zero$\left( 0 \right)$ .
$x = \dfrac{{ - 2k + 3}}{{k + 1}}$$ - - - - - - - - \left( 1 \right)$
And $y = 0 = \dfrac{{7k - 3}}{{k + 1}}$
Or $7k - 3 = 0$
Or $7k = 3$
Or $k = \dfrac{3}{7}$
Substituting the value of $k$ in equation $\left( 1 \right)$ ,
$x = \dfrac{{ - 2 \times \left( {\dfrac{3}{7}} \right) + 3 \times 1}}{{k + 1}}$
Or $x = \dfrac{{\dfrac{{ - 6}}{7} + 3}}{{\dfrac{3}{7} + 1}}$
Or $x = \dfrac{{\dfrac{{ - 6 + 21}}{7}}}{{\dfrac{{3 + 7}}{7}}}$
Or $x = \dfrac{{15}}{{10}}$
Thus , $x = \dfrac{3}{2}$
Hence, $P\left( {\dfrac{3}{2},0} \right)$

Note –Whenever we face this type of question the key concept is that. Simply , we have to apply Section formula as theses points are divided by $x - $ axis the $y$coordinates must be zero i.e. $P\left( {x,0} \right)$ .