Let a and b be any two numbers satisfying 1/a² + 1/b² = 1/4. Then, the foot of perpendicular from the origin on the variable line, xa + yb = 1, lies on:

Given equation of line is xa + yb = 1 ⇒ bx + ay - ab = 0 (1)
Slope of this line is -b/a
Slope of perpendicular line is a/b
Equation of perpendicular from origin is y = (a/b)x
Substituting in (1), we get bx + a(ax/b) - ab = 0
bx + a²x/b - ab = 0
b²x + a²x - ab² = 0
x(a² + b²) = ab²
x = ab²/(a² + b²)
Also, y = (a/b)x = a/b(ab²/(a² + b²)) = a²b/(a² + b²)
Let (h, k) be the foot of perpendicular
h = ab²/(a² + b²)
k = a²b/(a² + b²)
h² + k² = a²b⁴/(a² + b²)² + a⁴b²/(a² + b²)²
= a²b² (b² + a²)/(a² + b²)² = a²b²/(a² + b²)
1/a² + 1/b² = 1/4
(a² + b²)/a²b² = 1/4
a²b² = 4(a² + b²)
h² + k² = a²b²/4(a² + b²) = 1/4(a²b²/ (a² + b²)) = 1/4 (4) = 1
h² + k² = 1
Hence, the foot of perpendicular from the origin on the variable line lies on a circle with radius 1.