Let `A(k)` be the area bounded by the curves `y=x^2-3` and `y=k x+2` The range of `A(k)` is `((10sqrt(5))/3,oo)` The range of `A(k)` is `((20sqrt(5))/3,oo)` If function `kvecA(k)` is defined for `k in [-2,oo` ), then `A(k)` is many-one function. The value of `k` for which area is minimum is `1.`

Options

A. The range of A(k) is `[(10sqrt(5))/(3),oo)`
B. The range of A(k) is `[(20sqrt(5))/(3),oo)`
C. If function `krarrA(k)` is defined for `k in [-2,oo),` then A(k) is many-one function
D. The value of k for which area is minimum is 1

Answer

B, C

Explanation

Line `y=kx+2` passes through fixed point (0,2) for different value of k.
Also, it is obvious that minimum A(k) occurs when k=0, as when line is rotated from this position about point (0,2), the increased part of area is more than the decreased part of area. Therefore,
`"Minimum area "=2overset(sqrt(5))underset(0)int(2-(x^(2)-3))dx`
`=2overset(sqrt(5))underset(0)int(5-x^(2))dx`
`=2[5x-(x^(3))/(3)]_(0)^(sqrt(5))`
`=2[5sqrt(5)-(5sqrt(5))/(3)]`
`=(20sqrt(5))/(3)` sq. units