In what ratio does the x-axis divide the line segment joining the points (-1, -12) and (3, 4)?

Introduction / Context:
This coordinate geometry question asks you to find the ratio in which the x-axis divides the line segment joining two points, one below and one above the x-axis. The x-axis corresponds to all points with y = 0, so the point of intersection lies where the y coordinate of the segment becomes zero. This problem tests your understanding of section formula, linear interpolation and the idea that the line segment is divided in proportion to distances from the intercept point.

Given Data / Assumptions:

- Point A is (-1, -12). - Point B is (3, 4). - The x-axis is the line y = 0. - The x-axis intersects the segment AB at some point P where y = 0. - We need the ratio in which P divides AB, typically written as AP : PB.

Concept / Approach:
The line segment between A and B is divided by the x-axis at a point with y coordinate zero. Because the y coordinates change linearly along the segment, the distances from P to A and from P to B are proportional to the magnitudes of the y coordinates at A and B. Alternatively, we can use the section formula for internal division: if a point P divides the segment joining A(x1, y1) and B(x2, y2) in the ratio m : n, then its coordinates are ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n)). We use the condition that the y coordinate of P is zero to find the ratio m : n.

Step-by-Step Solution:
Step 1: Let A(-1, -12) and B(3, 4). Step 2: Suppose the x-axis (y = 0) cuts AB at point P, dividing AB in the ratio m : n, where AP : PB = m : n. Step 3: By the section formula, the y coordinate of P is (m * 4 + n * (-12)) / (m + n). Step 4: Since P lies on the x-axis, its y coordinate is 0. Therefore, (4m - 12n) / (m + n) = 0. Step 5: The numerator must be zero, so 4m - 12n = 0. Step 6: Divide both sides by 4 to get m - 3n = 0, so m = 3n. Step 7: Hence the ratio m : n is 3 : 1, meaning AP : PB = 3 : 1.

Verification / Alternative check:
You can also reason using vertical distances from the x-axis. Point A is 12 units below the x-axis (y = -12), and point B is 4 units above it (y = 4). The x-axis will divide the segment in the inverse ratio of these distances, giving AP : PB = 3 : 1 because 12 : 4 simplifies to 3 : 1. This matches the algebraic result from the section formula.

Why Other Options Are Wrong:
Ratios such as 1 : 3, 3 : 2 and 2 : 3 do not match the derived relationship m = 3n. For example, 1 : 3 would imply that AP is shorter than PB, which contradicts the fact that the point is closer in y value to B (4 units) than to A (12 units). Only 3 : 1 correctly reflects the distances to the x-axis and satisfies the section formula equation.

Common Pitfalls:
Some learners mistakenly invert the ratio, writing 1 : 3 instead of 3 : 1, or confuse AP : PB with BP : PA. Others may misapply the section formula by swapping x and y coordinates or by setting the wrong coordinate to zero. Being clear about which segment corresponds to which part of the ratio and carefully using the y coordinate to enforce y = 0 helps avoid these errors.

Final Answer:
The x-axis divides the line segment joining (-1, -12) and (3, 4) in the ratio 3 : 1, which corresponds to option C.