In coordinate geometry, point A(4, 2) divides the line segment BC internally in the ratio 2 : 3. Given that B has coordinates (2, 6) and C has coordinates (7, y), find the exact value of y.

Introduction / Context:
This question tests the concept of internal division of a line segment in coordinate geometry. A point that divides a segment in a given ratio can be expressed as a weighted average of the endpoints. Here, you are given the coordinates of two points B and A, the ratio in which A divides BC, and the x coordinate of C, and you must determine the unknown y coordinate of C. This is a standard application of the section formula or vector approach.

Given Data / Assumptions:

• Point B has coordinates (2, 6).
• Point C has coordinates (7, y), where y is unknown.
• Point A has coordinates (4, 2) and lies on segment BC.
• A divides BC internally in the ratio 2 : 3, meaning BA : AC = 2 : 3.
• All points lie in the same plane with standard Cartesian coordinates.

Concept / Approach:
If a point P divides the segment joining points B(x1, y1) and C(x2, y2) internally in the ratio m : n, where BP : PC = m : n, then its coordinates can be expressed using the section formula. One convenient vector based description uses P = B + (m / (m + n)) * (C − B). In this problem, A divides BC in the ratio BA : AC = 2 : 3, so A is located 2/5 of the way from B to C. We can use this to write the coordinates of A in terms of B and C and then solve for the unknown y coordinate.

Step-by-Step Solution:
1) Let B = (2, 6), C = (7, y), and A = (4, 2). 2) The ratio BA : AC is 2 : 3, so A is 2 parts along the segment from B to C out of a total of 5 parts. 3) Write the vector from B to C as C − B = (7 − 2, y − 6) = (5, y − 6). 4) Then A can be expressed as A = B + (2/5) * (C − B). 5) Compute A from this formula: A = (2, 6) + (2/5) * (5, y − 6) = (2 + 2/5 * 5, 6 + (2/5)(y − 6)). 6) Simplify the x coordinate: 2 + 2/5 * 5 = 2 + 2 = 4, which matches the given x coordinate of A and confirms our interpretation of the ratio. 7) For the y coordinate, we have 6 + (2/5)(y − 6) = 2. 8) Subtract 6 from both sides: (2/5)(y − 6) = 2 − 6 = −4. 9) Multiply both sides by 5/2: y − 6 = −4 * (5/2) = −10. 10) Add 6 to both sides: y = −10 + 6 = −4.

Verification / Alternative check:
Substitute y = −4 back into C and verify that A divides BC in the stated ratio. With C = (7, −4), the vector BC is (5, −10). The vector BA is A − B = (4 − 2, 2 − 6) = (2, −4). Notice that BA = (2/5) * BC, since (2, −4) = (2/5)(5, −10). This confirms that A is indeed located 2/5 of the way from B to C, corresponding to BA : AC = 2 : 3, so the computed y value is consistent.

Why Other Options Are Wrong:
Options b (6) and c (8) would place C above or to the same height as A in a way that does not satisfy the 2 : 3 ratio along the vertical direction. Options d (−8) and e (−6) produce different vertical displacements that do not make BA equal to 2/5 of BC. Only option a, y = −4, ensures that the coordinates satisfy the internal division ratio in both x and y directions.

Common Pitfalls:
One common mistake is mixing up the order of the ratio, interpreting 2 : 3 as AC : BA instead of BA : AC. Another is mis applying the section formula and using 3/5 instead of 2/5 as the weight from B towards C. Errors in basic algebra when solving the equation for y can also lead to incorrect answers. Re writing the internal division in vector form and checking both coordinates is a robust way to avoid these errors.

Final Answer:
The value of y that makes A(4, 2) divide BC in the ratio 2 : 3 is -4.