Let C be a circle with centre P0 and AB be the diameter of C. Suppose P1 is the mid-point of the line segment P0B,

Question

Let C be a circle with center P₀ and AB be the diameter of C. Suppose P₁ is the mid-point of the line segment P₀B, P₂ is the mid-point of the line segment P₁B and so on. Let C₁, C₂, C₃, ....... be the circles with diameters P₀P₁, P₁P₂, P₂P₃, ..... respectively. Suppose the circles C₁, C₂, C₃ ...... are all shaded. The ratio of the area of the unshaded portion of C to the original circle C is:

(a) 8: 9 

(b) 9: 10 

(c) 10: 11 

(d) 11: 12

Answer 1

Answer : (d) 11: 12

Area of C = πr², where P₀B = r is the radius of the circle C. 

Then radius of circle C₁ = r/4 

radius of circle C₂ = r/8 and so on. 

∴ Area of shaded portion = Area of C₁ + Area of C₂ + ........

= \(\frac{\pi r^2}{16}\) + \(\frac{\pi r^2}{64}\) + ... 

= πr² [ \(\frac{1}{16}\)+ \(\frac{1}{64}\)+ ... ] 

= 11: 12