Intersecting lines between tops and opposite feet of two poles: Two vertical poles of heights a and b meters stand p meters apart (b > a). Join the top of each pole to the foot of the other; at what height above the ground do these two lines intersect?

Introduction / Context:
Two straight lines join the top of one pole to the foot of the other, forming an X-shaped crossing. Coordinate geometry (or section formulas) finds the intersection height independent of the horizontal distance p.

Given Data / Assumptions:

• Pole 1 at x = 0 has top (0, a); foot (0, 0).
• Pole 2 at x = p has top (p, b); foot (p, 0).
• Lines: L1 from (0, a) to (p, 0); L2 from (0, 0) to (p, b).

Concept / Approach:
Parameterize both lines and solve for the intersection where x-coordinates match. Then read off the y-coordinate (height). Alternatively, use similar triangles/section ratios.

Step-by-Step Solution (parametric):
L1: (x, y) = (tp, a(1 − t)), 0 ≤ t ≤ 1.L2: (x, y) = (sp, sb), 0 ≤ s ≤ 1.At intersection, tp = sp ⇒ t = s.Also, a(1 − t) = sb ⇒ sb + at = a ⇒ t(same) = a/(a + b).Then y = sb = b·(a/(a + b)) = ab/(a + b).

Verification / Alternative check:
By similar triangles along each line, the same ratio emerges; the result is independent of p.

Why Other Options Are Wrong:
Expressions like (a + b)/ab invert dimensions; p appears in some options but cancels out in the correct derivation.

Common Pitfalls:
Forgetting that both lines share the same division ratio along the horizontal span; mishandling parameters.

Final Answer:
ab/(a + b)