Question

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.

• A. $(\frac{-34}{13},0)$.
• B. $(\frac{-30}{13},0)$.
• C. $(\frac{-34}{12},1)$.
• D. $(\frac{-32}{15},0)$.

Answer 1

The correct option is A

$(\frac{-34}{13},0)$.

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, $x_1=-4,x_2=-1$ and $y_1=-6,y_2=7$

$i.e.,\{\frac{\lambda (-1)+1(-4)}{\lambda +1},\frac{\lambda \left(7\right)+1(-6)}{\lambda +1}\}=(\frac{-\lambda -4}{\lambda +1},\frac{7\lambda -6}{\lambda +1})$

But according to the question, line segment joining A(-4,-6) and B(-1,7) is divided by the X-axis.

So, y-coordinate must be zero.

$\therefore \frac{7\lambda -6}{\lambda +1}=0\Rightarrow 7\lambda -6=0$

$\therefore \lambda =\frac{6}{7}$

So, the required ratio 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., $(\frac{\frac{-34}{7}}{\frac{13}{7}},\frac{\frac{6-6}{13}}{\frac{13}{7}})i.e.,(\frac{-34}{13},0)$

Hence, the required point of division is $(\frac{-34}{13},0)$.