Point A(2, 1) divides the line segment joining points B(1, -3) and C(4, y) internally in the ratio 2 : 3. What is the value of y, the y-coordinate of point C?

Introduction / Context:
This problem uses the section formula in coordinate geometry. A point divides a line segment joining two other points in a given ratio, and we use that information to find an unknown coordinate of one endpoint. Such questions reinforce understanding of internal division of line segments in the Cartesian plane.

Given Data / Assumptions:

• Point B has coordinates (1, -3).
• Point C has coordinates (4, y), with y unknown.
• Point A has coordinates (2, 1).
• Point A divides segment BC internally in the ratio 2 : 3.

Concept / Approach:
If a point divides the segment joining B(x1, y1) and C(x2, y2) internally in the ratio m : n (that is, BA : AC = m : n), then its coordinates are given by:
A(x, y) = ((n * x1 + m * x2) / (m + n), (n * y1 + m * y2) / (m + n)).We apply this formula to the y coordinates to find the unknown y for point C, using the fact that we already know the y coordinate of A.

Step-by-Step Solution:
Step 1: Let B(1, -3) be (x1, y1) and C(4, y) be (x2, y2).Step 2: The ratio in which A divides BC is 2 : 3, so m = 2 and n = 3.Step 3: For the y coordinate, apply the section formula to get y_A = (n * y1 + m * y2) / (m + n).Step 4: Substitute values: 1 = (3 * (-3) + 2 * y) / (2 + 3).Step 5: Simplify: 1 = (-9 + 2y) / 5.Step 6: Multiply both sides by 5: 5 = -9 + 2y, so 2y = 14 and y = 7.

Verification / Alternative check:
Using y = 7, the midpoint formula for the y coordinate with weighted ratio gives y_A = (-9 + 14) / 5 = 5 / 5 = 1, which matches the given y coordinate of A. This confirms that y = 7 is consistent with the given ratio and point A coordinates.

Why Other Options Are Wrong:
If y were 8, -7, -8 or 5, the computed y coordinate for point A using the section formula would not be 1, and so those values would not satisfy the given condition that A divides BC in the specified ratio. Only y = 7 makes the formula for internal division hold true.

Common Pitfalls:
Common mistakes include reversing the ratio m : n in the formula or mixing up the roles of B and C. Another frequent error is applying the section formula correctly to x but forgetting to apply it separately to y. Carefully writing the formula and substituting values step by step avoids these issues.

Final Answer:
The value of the y coordinate of point C is 7.