Find the ratio in which the line segment joining A(1, −5) and B(−4, 5) is divided by the x-axis. Also find the coordinates of the point of division.
Ratio = 1 : 1
Point of division = \(\left(-\dfrac{3}{2}, 0\right)\)
The point dividing the line segment AB and lying on the x-axis must have y–coordinate 0. Let this point be \((x, 0)\).
Points A and B are \(A(1, -5)\) and \(B(-4, 5)\).
The y–coordinate of a point dividing AB in the ratio \(m : n\) is given by the section formula: \( y = \frac{m y_B + n y_A}{m + n} \).
Since the point lies on the x-axis, its y–coordinate is 0. So we set
\( 0 = \frac{m(5) + n(-5)}{m + n}. \)
Multiplying both sides by \(m + n\), we get: \( 5m - 5n = 0. \)
Simplifying: \( 5m = 5n \Rightarrow m = n. \)
Thus, the required ratio is \(1 : 1\).
Now we find the coordinates of the midpoint (since the ratio is 1:1). Using the midpoint formula:
\( x = \frac{1 + (-4)}{2} = \frac{-3}{2}, \qquad y = \frac{-5 + 5}{2} = 0. \)
Therefore, the point of division is \(\left( -\frac{3}{2}, 0 \right)\).