Question

A straight rod of length 8 units slides with its ends A and B always on the x and y axes respectively. Find the locus of the midpoint of the line segment AB

Answer 1

Given A and B are the ends of the straight rod of length 8 unit on the x and y-axes.

Let A be (a, 0) and B(0, b).

Let M(h, k) be the midpoint of AB(h, k)

= (h, k) = `(("a" + 0)/2, (0 +"b")/2)`

= `("a"/2, "b"/2)`

h = `"a"/2`

k = `"b"/2`

⇒ a = 2h, b = 2k

In the right-angled ∆OAB

AB² = OA² + OB²

8² = a² + b²

64 = (2h)² + (2k)

64 = 4h² + 4k² 

h² + k² = `64/4` = 16

The locus of M (h, k) is obtained by replacing h by x and k by y

∴ The required locus is x² + y² = 16