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 m1:m2.
Step 2: Apply the section formula for y-coordinate
We use the section formula for the y -coordinate. Here, (x1,y1)=(1,−5) and (x2,y2)=(−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=k+15k−5 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, (x1,y1)=(1,−5) and (x2,y2)=(−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 (−23,0).