Question
Question: The point ([P + 1], [P]) (where [x] is the greatest integer less than or equal to x), lying inside t...
The point ([P + 1], [P]) (where [x] is the greatest integer less than or equal to x), lying inside the region bounded by the circle x2 + y2 –2x –15 = 0 and x2 + y2 –2x – 7 = 0. Then
A
PĪ [1, 0) Č (0, 1) Č (1, 2)
B
P Ī (0, 1)
C
P Ī (1, 2)
D
None of these
Answer
None of these
Explanation
Solution
Since the ([P +1], [P]) lies inside the circle
x2 + y2 –2x –15 = 0
\ ([P + 1]2 + [P]2 –2 ([P + 1]) –15 < 0
([P] + 1)2 + [P]2 –2 ([P] + 1) – 15 < 0
[Q [x + n] = [x] + n, n Ī I]
[P]2 + 1 + 2[P] + [P]2 –2[P] –2 –15 < 0,
Ž 2[P]2 –16 < 0, [P]2 < 8 ... (i)
From the second circle ([P] + 1)2 + [P]2 – 2
([P[ + 1) – 7 > 0
Ž 2[P]2 – 8 > 0, [P]2 > 4 ... (ii)
From (i) and (ii), 4 < [P]2 < 8, which is not possible.
\ for no values of 'P' the point will be within the region.