The point P(5, -2) divides the segment joining A(x, 0) and B(0, y) internally in the ratio 2 : 5. Using the section formula for internal division, find the exact values of x and y.

Introduction / Context:
This coordinate geometry question tests the section formula (also called the internal division formula). When a point divides a line segment joining two points in a given ratio m:n internally, its coordinates are a weighted average of the endpoints. Here, the endpoints are on the axes: A is on the X-axis at (x, 0) and B is on the Y-axis at (0, y). The dividing point P is given as (5, -2). Using the internal section formula separately for x-coordinate and y-coordinate gives two equations, allowing us to solve for x and y exactly.

Given Data / Assumptions:

• A = (x, 0) • B = (0, y) • P = (5, -2) • P divides AB internally in ratio 2 : 5 (AP : PB = 2 : 5) • Required: values of x and y

Concept / Approach:
Internal division (section) formula: If P divides A(x1, y1) and B(x2, y2) in ratio m:n internally, then: Px = (m*x2 + n*x1) / (m + n) Py = (m*y2 + n*y1) / (m + n) Here m = 2, n = 5, A = (x, 0) and B = (0, y). Substitute and solve the two equations: 5 = (2*0 + 5*x)/7 and -2 = (2*y + 5*0)/7.

Step-by-Step Solution:
1) Identify values: A(x1, y1) = (x, 0), B(x2, y2) = (0, y), m = 2, n = 5 2) Apply section formula for x-coordinate: Px = (m*x2 + n*x1)/(m + n) = (2*0 + 5*x)/7 = 5 3) Solve for x: (5x)/7 = 5 5x = 35 x = 7 4) Apply section formula for y-coordinate: Py = (m*y2 + n*y1)/(m + n) = (2*y + 5*0)/7 = -2 5) Solve for y: (2y)/7 = -2 2y = -14 y = -7 6) Therefore: x = 7 and y = -7

Verification / Alternative check:
With x = 7 and y = -7: A = (7, 0), B = (0, -7). Compute P using formula: Px = (2*0 + 5*7)/7 = 35/7 = 5, Py = (2*(-7) + 5*0)/7 = -14/7 = -2. It matches P(5, -2), confirming correctness.

Why Other Options Are Wrong:
• Any option with x = -7 would give Px = (5*(-7))/7 = -5, not +5. • Any option with y = +7 would give Py = (2*(7))/7 = 2, not -2. • x = ±3 values fail the x-coordinate equation immediately.

Common Pitfalls:
• Reversing the weights m and n in the section formula. • Using external division formula by mistake (signs change there). • Forgetting that A is (x, 0) and B is (0, y), so one coordinate is already zero at each endpoint.

Final Answer:
x = 7; y = -7