In coordinate geometry, in what ratio does the point T(x, 0) on the x-axis divide the segment joining the points S(-4, -1) and U(1, 4)?

Introduction / Context:
This problem uses the concept of internal division of a line segment in coordinate geometry. A point T lies on the x-axis and divides the line segment joining two given points S and U. Because T is on the x-axis, its y-coordinate is zero, which allows us to use the section concept or a simple linear interpolation approach on the y-coordinates to find the ratio in which T divides the segment. Such questions frequently appear in aptitude and entrance exams to test understanding of coordinate geometry basics.

Given Data / Assumptions:
• Point S has coordinates S(-4, -1). • Point U has coordinates U(1, 4). • Point T lies on the x-axis, so T has coordinates (x, 0). • T divides segment SU internally in some ratio k : (1 − k) or m : n which we must find.

Concept / Approach:
If a point divides a line segment internally in a given ratio, its coordinates can be obtained using the section formula. However, in this question, a more direct way is to focus on the y-coordinates. The y-coordinate varies linearly between the endpoints of the segment. Because T is on the x-axis, its y-coordinate is zero, so we can write an equation based on the change from -1 to 4. The parameter that takes us from -1 to 0 will give the fraction of the segment from S to T. From that fraction we can deduce the ratio ST : TU.

Step-by-Step Solution:
Step 1: Let T divide the segment from S to U in the ratio ST : TU = k : (1 − k), where k is the fraction from S. Step 2: Parameterize the y-coordinate along SU. Starting from S, the y-coordinate after fraction k is y = -1 + k * (4 - (-1)). Step 3: Simplify the change: 4 - (-1) = 5. So y = -1 + 5k. Step 4: Since T is on the x-axis, its y-coordinate must be zero. So set -1 + 5k = 0. Step 5: Solve for k: 5k = 1, so k = 1/5. Step 6: This means ST : SU = 1 : 5, so ST : TU = (1/5) : (4/5) = 1 : 4.

Verification / Alternative Check:
We can find the coordinates of T explicitly. For the x-coordinate, x = -4 + k * (1 - (-4)) = -4 + (1/5) * 5 = -4 + 1 = -3. So T(-3, 0). Now compute distances ST and TU using rough comparisons or actual lengths. T is closer to S than to U, and the ratio of the progress along the segment based on the parameter is indeed 1 : 4. This matches our earlier result.

Why Other Options Are Wrong:
• 4 : 1 would imply T is much closer to U than to S, which is not consistent with the y-interpolation that gives k = 1/5. • 1 : 2 and 2 : 1 are intermediate ratios and do not satisfy the condition y = 0 when we interpolate between -1 and 4.

Common Pitfalls:
Students sometimes try to work with the x-coordinates first, which can be more confusing. Another frequent mistake is to assume that equal changes in x and y correspond to equal ratios, which is not correct. The safest approach is to use the parameter on the y-coordinate because the condition y = 0 is very clear and leads directly to the correct ratio.

Final Answer:
The point T divides the segment joining S(-4, -1) and U(1, 4) in the ratio 1 : 4 (ST : TU).