A point T with coordinates (x, 0) lies on the line segment joining points S(-4, -1) and U(1, 4). In what ratio does point T divide the segment SU?

Introduction / Context:
This analytic geometry question deals with internal division of a line segment and uses the section formula. It also adds the condition that the dividing point lies on the x axis, which allows us to use the y coordinate to determine the ratio in which the segment is divided.

Given Data / Assumptions:

• Point S has coordinates (-4, -1).
• Point U has coordinates (1, 4).
• Point T = (x, 0) lies on segment SU.
• T divides SU in some ratio k : 1 or 1 : k that we need to find.
• Division is assumed to be internal.

Concept / Approach:
A point dividing a segment internally in a given ratio can be found using the section formula. However, here we are given that T lies on the x axis (its y coordinate is zero). We can parameterise the segment using a parameter t, or directly use the section formula in reverse by relating the y coordinate of T to the ratio of division.

Step-by-Step Solution:
Step 1: Let S(-4, -1) and U(1, 4). Let T divide SU in the ratio ST : TU = m : n. Step 2: By the section formula, the y coordinate of T is given by (n * y_S + m * y_U) / (m + n). Step 3: So y_T = (n * (-1) + m * 4) / (m + n). Step 4: We are told T lies on the x axis, so y_T = 0. Step 5: Set the expression equal to 0: ( -n + 4m ) / (m + n) = 0. Step 6: For a fraction to be zero, its numerator must be zero. So -n + 4m = 0. Step 7: Rearranging gives 4m = n, hence n : m = 4 : 1, or m : n = 1 : 4. Step 8: Therefore ST : TU = 1 : 4.

Verification / Alternative check:
We can use a parameter t representing the fraction of the way from S to U. Coordinates of a point on SU are S + t(U - S) = (-4 + 5t, -1 + 5t). Set y = 0: -1 + 5t = 0, so t = 0.2. Hence ST : TU = 0.2 : 0.8 = 1 : 4, confirming the result.

Why Other Options Are Wrong:
4 : 1: This is the reverse of the correct ratio and corresponds to a point closer to U than to S.
1 : 2 and 2 : 1: These give different y coordinates and do not result in y = 0 for point T.
3 : 1: Also leads to a non zero y coordinate, so T would not lie on the x axis.

Common Pitfalls:
A frequent mistake is misusing the section formula or mixing up which coefficient multiplies which point. Another error is forgetting that the denominator of the section formula does not affect the zero value, and only the numerator must be set to zero. Careful algebra and clear assignment of the ratio help avoid these issues.

Final Answer:
Point T divides segment SU in the ratio 1 : 4.