Point A divides the line segment BC internally in the ratio 1 : 3. 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 is an internal division problem in coordinate geometry. It uses the section formula, which provides coordinates of a point that divides a line segment joining two given points in a specified ratio. This formula is widely used in analytic geometry and vector problems.

Given Data / Assumptions:

• Point B has coordinates (4, -4).
• Point C has coordinates (0, 6).
• Point A divides BC in the ratio 1 : 3.
• We interpret the ratio as BA : AC = 1 : 3 (internal division).
• We must find coordinates of A.

Concept / Approach:
For internal division, if A divides BC in the ratio m : n (BA : AC = m : n), and B(x1, y1), C(x2, y2), then: A(x, y) = ((n x1 + m x2) / (m + n), (n y1 + m y2) / (m + n)) This is the section formula for internal division.

Step-by-Step Solution:
Here, B(4, -4), C(0, 6), and BA : AC = 1 : 3 ⇒ m = 1, n = 3. Apply the formula for x-coordinate: x = (n x1 + m x2) / (m + n) = (3 * 4 + 1 * 0) / (1 + 3). x = (12 + 0) / 4 = 12 / 4 = 3. Now for y-coordinate: y = (n y1 + m y2) / (m + n) = (3 * (-4) + 1 * 6) / 4. y = (-12 + 6) / 4 = -6 / 4 = -1.5. Therefore A has coordinates (3, -1.5).

Verification / Alternative check:
We can verify that A is closer to B than to C because the ratio is 1 : 3. B is (4, -4) and C is (0, 6). The x-coordinate 3 is nearer to 4 than to 0, and y-coordinate -1.5 is between -4 and 6, but much closer to -4, consistent with a 1 : 3 division from B to C.

Why Other Options Are Wrong:
(-3, 1.5) and (-1.5, 3) place A on the opposite side of C or in locations inconsistent with the line segment direction. (1.5, 3) would correspond to a different ratio, not the specified 1 : 3 from B to C.

Common Pitfalls:
Mixing up m and n, or mistakenly using m with B and n with C in the wrong positions, often leads to incorrect coordinates. Another common issue is misinterpreting the ratio as AC : BA instead of BA : AC. Carefully define which part of the segment each ratio component refers to before applying the formula.

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