Point A divides the line segment BC internally in the ratio 5 : 1. The coordinates of B are (6, −4) and the coordinates of C are (0, 8). What are the coordinates of point A?

Introduction / Context:
This is a coordinate geometry question involving internal division of a line segment. It tests your knowledge of the section formula, which gives the coordinates of a point dividing a segment in a given ratio.

Given Data / Assumptions:

• Segment BC has endpoints B(6, −4) and C(0, 8).
• Point A divides BC internally in the ratio 5 : 1.
• The ratio 5 : 1 means BA : AC = 5 : 1, so A is closer to C because AC is the smaller part.
• We must find the coordinates (x, y) of point A.

Concept / Approach:
For a segment joining points P(x1, y1) and Q(x2, y2), if a point R divides PQ internally in the ratio m : n, where PR : RQ = m : n, then the coordinates of R are given by the section formula: R = ((n x1 + m x2) / (m + n), (n y1 + m y2) / (m + n)) Here, P corresponds to B, Q corresponds to C, and R corresponds to A, with BA : AC = 5 : 1, so m = 5 and n = 1.

Step-by-Step Solution:
Step 1: Let B(x1, y1) = (6, −4) and C(x2, y2) = (0, 8). Step 2: The ratio BA : AC is 5 : 1, so m = 5 and n = 1 in the section formula. Step 3: Using the formula for internal division, coordinates of A are: A = ((n x1 + m x2) / (m + n), (n y1 + m y2) / (m + n)). Step 4: Substitute x1 = 6, y1 = −4, x2 = 0, y2 = 8, m = 5 and n = 1. Step 5: x coordinate of A: xA = (1 * 6 + 5 * 0) / (5 + 1) = 6 / 6 = 1. Step 6: y coordinate of A: yA = (1 * (−4) + 5 * 8) / 6 = (−4 + 40) / 6 = 36 / 6 = 6. Step 7: Therefore, A has coordinates (1, 6).

Verification / Alternative check:
We can confirm that A is closer to C than to B. The vertical difference from C(0, 8) to A(1, 6) is only 2 units in y and 1 unit in x, while B(6, −4) is much farther away. Also, the direction from B to C roughly moves from (6, −4) to (0, 8), and A(1, 6) lies along that line segment, which visually supports the calculation.

Why Other Options Are Wrong:
(-1, 6) and (-1, -6): These place A on the left side of both B and C and do not lie on the segment connecting B and C correctly for the given ratio.
(1, -6): This point has the correct x coordinate from the calculation but a completely incorrect y coordinate and lies outside the segment.
(2, 4): This point is on the line passing through B and C but corresponds to a different division ratio, not 5 : 1.

Common Pitfalls:
Students sometimes reverse the ratio in the section formula or mistakenly use m x1 + n x2 instead of n x1 + m x2. Another error is to treat the ratio as a fraction of total distance without using coordinates properly. Carefully labeling which part corresponds to which weight in the formula helps avoid these mistakes.

Final Answer:
The coordinates of point A are (1, 6).