Consider two circles `C_1: x^2+y^2-1=0` and `C_2: x^2+y^2-2=0`. Let A(1,0) be a fixed point on the circle `C_1` and B be any variable point on the circle `C_2`. The line BA meets the curve `C_2` again at C. Which of the following alternative(s) is/are correct?

Options

A. `OA^(2) +OB^(2) +BC^(2) in [7,11],` where O is the origin
B. `OA^(2)+OB^(2) +BC^(2) in [4,7]`, where O is the origin
C. Locus of midpoint of AB is a circle of radius `(1)/(sqrt(2))`
D. Locus of midpoint of AB is a circle of area `(pi)/(2)`

Answer

A, C

Explanation

We have maximum `BC = 2sqrt(2)` and minimum `BC =2`.
`:. OA^(2) +OB^(2) +BC^(2) in [7,11]`

Let M be the mid-point of AB.
`rArr M -= ((1+sqrt(2)cos theta)/(2),(sqrt(2)sintheta)/(2))-=(h,k)`
`:. sin theta = (2k)/(sqrt(2)), cos theta = (2h-1)/(sqrt(2))`
Now, on squaring and adding we get
`rArr 4k^(2) +(2h-1)^(2) =2`
`:.` LOcus of `M(h,k): (x-(1)/(2))^(2) +y^(2) = ((1)/(sqrt(2)))^(2)`