Point A divides segment BC in the ratio 1 : 3 internally. If the coordinates of B are (4, -4) and the coordinates of C are (0, 6), what are the coordinates of point A?

Introduction / Context:
This question is from coordinate geometry and uses the section formula, which gives the coordinates of a point that divides a line segment between two given points in a specified ratio. Problems like this appear frequently in analytic geometry and help you understand how weighted averages of coordinates describe positions on line segments. Here, point A divides segment BC internally in the ratio 1 : 3, and we must compute its coordinates.

Given Data / Assumptions:

• Point B has coordinates (4, -4).
• Point C has coordinates (0, 6).
• Point A divides segment BC in the ratio 1 : 3 internally.
• By convention, this means BA : AC = 1 : 3, so A is closer to B than to C.
• We must find the coordinates (x, y) of point A.

Concept / Approach:
The internal section formula for a point A dividing segment BC in the ratio m : n (where BA : AC = m : n) is:
Coordinates of A = ( (n * x_B + m * x_C) / (m + n), (n * y_B + m * y_C) / (m + n) ).
This formula arises from considering A as a weighted average of B and C, with weights proportional to the lengths of the opposite segments. We will assign m = 1 and n = 3, substitute the coordinates of B and C, and simplify to find the coordinates of A.

Step-by-Step Solution:
Let B(4, -4) and C(0, 6).Given BA : AC = 1 : 3, so m = 1 and n = 3.Using the section formula for internal division:x-coordinate of A = (n * x_B + m * x_C) / (m + n) = (3 * 4 + 1 * 0) / (1 + 3) = 12 / 4 = 3.y-coordinate of A = (n * y_B + m * y_C) / (m + n) = (3 * (-4) + 1 * 6) / 4 = (-12 + 6) / 4 = -6 / 4 = -1.5.Therefore, A has coordinates (3, -1.5).

Verification / Alternative check:
To check the ratio visually, compute the distance along the x axis and y axis from B to A and from A to C. From B(4, -4) to A(3, -1.5), the change is (-1, 2.5). From A(3, -1.5) to C(0, 6), the change is (-3, 7.5). Each component change from A to C is exactly three times the corresponding change from B to A, matching the ratio 1 : 3. This confirms that A lies on segment BC and divides it in the correct ratio.

Why Other Options Are Wrong:
Option (-3, 1.5) lies to the left of both B and C and does not lie on segment BC. Option (-1.5, 3) similarly does not divide the segment between B and C in the stated ratio. Option (1.5, 3) and (2, 1) may lie near the line between B and C but they do not satisfy the 1 : 3 internal division when tested using the section formula or distance ratios. Only (3, -1.5) correctly divides BC in the ratio 1 : 3.

Common Pitfalls:
Students often mix up which weight multiplies which endpoint. Remember that in the internal section formula for BA : AC = m : n, the coordinate of A uses n with B and m with C. Another frequent error is to assume that the ratio describes BC : AC instead of BA : AC, leading to reversed weights. Keeping the ratio interpretation consistent and writing the formula clearly avoids these mistakes.

Final Answer:
The coordinates of point A are (3, -1.5).