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From above expression we get

Hence the region bouded by the curve and y = 0 ia semi-circle centered at origin

and having radius 5 as showm in figure +.

Let P be a point with coordinates ( a, a+1) in the region. Then we have

a² + (a+1)²  ≤  25

2a² + 2a -24 ≤ 0

a² + a - 12 ≤ 0

From above expression , we get a ≤ 3 . 

But  a+1 > 0 because y-coordinate of point P can not be negative.

Hence a lies between -1 and 3 ,  -1 ≤ a ≤ 3

Hence the answer  :-  a ∈ (-1 , 3 )