In coordinate geometry, in what ratio does the point T(x, 0) on the x-axis divide the segment joining the points S(5, 1) and U(-1, -2)?

Introduction / Context:
This is another coordinate geometry problem involving internal division of a line segment. The segment joins two points S and U in the plane, and a point T on the x-axis divides this segment. Because T lies on the x-axis, its y-coordinate is zero, which gives a direct equation involving the y-coordinates of S and U. From that, we can find in what ratio T divides the segment SU. This type of question tests understanding of linear interpolation and section concepts in a simple but elegant way.

Given Data / Assumptions:
• Point S has coordinates S(5, 1). • Point U has coordinates U(-1, -2). • Point T lies on the x-axis, so T has coordinates (x, 0). • T divides SU internally in some ratio which we must determine.

Concept / Approach:
If a point T divides a line segment between S and U in a certain ratio, the coordinates of T can be expressed as a weighted average of the coordinates of S and U. However, here we do not actually need to compute the x-coordinate. It is easier to work with the y-coordinate because we know the y-coordinate of T is zero. We treat movement along the segment as a linear change from the y-coordinate of S to the y-coordinate of U and set up an equation based on a fraction parameter. The resulting fraction directly gives the ratio in which T divides the segment.

Step-by-Step Solution:
Step 1: Let the fraction from S to T along SU be k, so that T corresponds to k of the way from S to U. Step 2: The y-coordinate along the segment is given by y = 1 + k * (-2 - 1). Step 3: Simplify the change in y: -2 - 1 = -3, so y = 1 - 3k. Step 4: Since T lies on the x-axis, its y-coordinate is zero, so set 1 - 3k = 0. Step 5: Solve for k: 3k = 1 gives k = 1/3. Step 6: Therefore ST : SU = 1 : 3, and TU : SU = 2 : 3. Hence ST : TU = (1/3) : (2/3) = 1 : 2.

Verification / Alternative Check:
We can also compute the coordinates of T explicitly. For the x-coordinate, x = 5 + (1/3) * (-1 - 5) = 5 + (1/3) * (-6) = 5 - 2 = 3. So T(3, 0). It is then clear that T lies between S and U, and the parameter k = 1/3 means that T is one third of the way from S towards U, leaving two thirds towards U. This confirms that the ratio ST : TU is indeed 1 : 2.

Why Other Options Are Wrong:
• 2 : 1 would mean T is much nearer to U, not supported by the calculation that gives k = 1/3. • 3 : 1 and 2 : 3 also do not match the fraction 1/3 for the distance from S to T.

Common Pitfalls:
Some students reverse the ratio and write TU : ST instead of ST : TU. Others incorrectly try to average the coordinates without using the idea of fractions along the segment. It is very important to define the parameter carefully and stick to a consistent interpretation of ST : TU versus TU : ST. Using the y-coordinate condition y = 0 is the cleanest and safest approach.

Final Answer:
The point T divides the segment joining S(5, 1) and U(-1, -2) in the ratio 1 : 2 (ST : TU).