Find the ratio in which the line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis, and find the coordinates of the point of division.

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Step-by-Step Solution

Step 1: Define the point of division and ratio

Since the point lies on the $x$-axis, its $y$-coordinate must be 0. We assume the ratio in which the line segment is divided is $k:1$. This simplifies calculations compared to $m_1:m_2$.

Step 2: Apply the section formula for y-coordinate

We use the section formula for the $y$-coordinate. Here, $(x_1, y_1) = (1, -5)$ and $(x_2, y_2) = (-4, 5)$. The $y$-coordinate of the point of division $P$ is 0.

Step 3: Solve for the ratio k

Substitute the known values into the section formula for $y$. Since $y=0$, we can solve the equation $0 = \frac{5k - 5}{k+1}$ for $k$. This gives $5k - 5 = 0$, which simplifies to $k=1$.

Step 4: State the ratio

Since $k=1$, the ratio $k:1$ becomes $1:1$. This means the $x$-axis divides the line segment AB in the ratio $1:1$, implying it is the midpoint.

Step 5: Apply the section formula for x-coordinate

Now we use the section formula for the $x$-coordinate with $k=1$. Here, $(x_1, y_1) = (1, -5)$ and $(x_2, y_2) = (-4, 5)$.

Step 6: Solve for x and state the coordinates

Substitute $k=1$ and the $x$-coordinates into the formula to find $x$. The coordinates of the point of division are $(x, 0)$, which is $(-\frac{3}{2}, 0)$.