Please give detailed explanation. Choose correct option.

Solution:

Let , CD  =  a  and given AB  =  2 CD , So

AB  =  2 a

And

Radius of given circle  =  r  , then AD  =  2r ( As AD is perpendicular to AB and CD )

Now we form our diagram , As :


Here , A be at origin and AB and AD at x - axis and y - axis respectively .

So,

Coordinates of A ( 0 , 0 ) , B ( 2a  , 0 ) , C ( a  , 2r )  and D ( 2r , 0 )

We can see ABCD is a trapezium , and we know area of trapezium  = Sum of parallel sides2×Height

So,

Area of ABCD  = AB  + CD2×AD

2a  + a2×2r = 183ar= 18ar= 183ar=6                                  --- ( 1 )

Now we draw a right angle triangle OBC and a perpendicular line from center to BC , As :

So,

tan θ = a - rr                                   --- ( 2 )           And tan  90° - θ = 2a - rr Cot θ = 2a - rr                                  (we know tan  90° - θ = Cot θ  )1 tan θ= 2a - rr                                  (we know Cot θ =1 tan θ )tan θ= r2a - r                                   --- ( 3 )   From equation 2 and 3 we get a - rr   = r2a - ra - r2a - r = r2 2a2 - 2ar - ar + r2 = r2 2a2 -3ar =0 a  2a -3 r =0 2a -3 r=0 2a =3ra =3r2 , Substitute that value in equation 1 and get 3r2× r = 63r22= 6r2= 6 ×23r2=2 ×2r2=4r =4r = ± 2
But radius can't be negative , So we get

Radius  =  r   = 2

Option B is correct

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