A region in the `x-y` plane is bounded by the curve `y=sqrt(25-x^2)` and the line `y=0` . If the point `(a ,a+1)` lies in the interior of the region, then (a)`a in (-4,3)` (b) `a in (-oo,-1)` ∪ `(3,oo)` (c)`a in (-1,3)` (d) none of these

Options

A. `a in (-4,3)`
B. `a in (- oo,-1) in (3, oo)`
C. `a in (-1),3)`
D. none of these

Answer

3

Explanation

`y=sqrt(25-x^(2))` and `y=0` bound the semicircle above the x-axis .
Therefore,
`a+1gt0` (1)
and `a^(2)+(a+1)^(2)-25lt0` or `2a^(2)+2a-24lt0`
or `a^(2)+a-12 lt0`
or `-4 lta lt3` (2)
From (1) and (2),
`-1 lta lt3`