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.

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 $(-\frac{3}{2}, 0)$.

More from this chapter

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• Find the coordinates of the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1 internally.
• Find the relation between x and y if the point (x, y) is equidistant from (7, 1) and (3, 5).
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• Find the coordinates of the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4).
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