Question

Question 10
In what ratio does the X-axis divide the line segment joining the points (-4,-6) and (-1,7)? Find the coordinates of the points of division.

Answer 1

Let the required ratio be $\lambda :1$ So, the coordinates of the point M of division A(-4,-6) and B(-1,7) are
$(\frac{\lambda x_2+1.x_1}{\lambda +1},\frac{\lambda y_2+1.y_1}{\lambda +1})$
Here, $x_1=-4,x_2=-1and{y}_1=-6,y_2=7$
$i.e.,\{\frac{\lambda (-1)+1(-4)}{\lambda +1},\frac{\lambda \left(7\right)+1(-6)}{\lambda +1}\}=(\frac{-\lambda -4}{\lambda +1},\frac{7\lambda -6}{\lambda +1})\text{but according to the question, line segment joining A(-4,-6)}\text{and B(-1,7) is divided by the X-axis.}\text{So, y-coordinate must be zero.}\therefore \frac{7\lambda -6}{\lambda +1}=0\Rightarrow 7\lambda -6=0\therefore \lambda =\frac{6}{7}\text{So, the required is 6:7 and the point of division M is}\{\frac{-\frac{6}{7}-4}{\frac{6}{7}+1},\frac{7\times \frac{6}{7}-6}{\frac{6}{7}+1}\}i.e.,(\frac{\frac{-37}{7}}{\frac{13}{7}},\frac{\frac{6-16}{13}}{\frac{13}{7}})i.e.,(\frac{-34}{13},0)\text{Hence, the required point of division is}(\frac{-34}{13},0)$.