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
P Ī [–1, 0) Č [0, 1) Č [1, 2)
B
P Ī [–1, 2) – {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
[But [x + n] = [x] + n, n Ī N]
\ [P + 1]2 + [P]2 – 2[P + 1] –15 < 0
([P] + 1)2 + [P]2 – 2([P] + 1) – 15 < 0
[P]2 + 1 + 2[P] + [P]2 – 2[P] – 2 – 15 < 0,
2[P]2 – 16 < 0, [P]2 < 8 … (1)
From the second circle
([P] + 1)2 + [P]2 –2([P] + 1) – 7 > 0
Ž 2[P]2 – 8 > 0, [P]2 > 4 … (2)
From (1) & (2), 4 < [P]2 < 8, which is not possible region.
\ For no values of ‘P’ the point will be within the region.