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.

Explanation

Let the required ratio be `lambda :1`. So , the coordinates of the point M of division A (-4,-6) and B(-1,7) are
`{(lambda x_(2)+1*x_(1))/(lambda+1),(lambda y_(2)+1*y_(1))/(lambda+1)}`
Here ,`x_(1)=-4,x_(2)=-1 and y_(1)=-6,y_(2)=7`
i.e., `((lambda(-1)+1(-4))/(lambda+1),(lambda(7)+1*(-6))/(lambda+1))=((-lambda-4)/(lambda+1),(7lambda-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.
`:. (7lambda-6)/(lambda+1)rArr7lambda-6=0`
`:. lambda=(6)/(7)`
So , the required ratio is `6 : 7` and the point of division M is `{(-(6)/(7)-4)/((6)/(7)+1),(7xx(6)/(7)-6)/((6)/(7)+1)}`
i.e., `((-34/(17))/(13/(7)),(6-6)/(13/(7)))` i.e., `((-34)/(13),0)`.
Hence , the required point of division is `((-34)/(13),0)`.