AB is a chord of the circle `x^2 + y^2 = 25/2` .P is a point such that PA = 4, PB = 3. If AB = 5, then distance of P from origin can be:

Options

A. `(9)/(sqrt(2))`
B. `(3)/(sqrt(2))`
C. `(5)/(sqrt(2))`
D. `(7)/(sqrt(2))` or `(1)/(sqrt(2))`

Answer

D

Explanation

`OA = OB =` radius `= (5)/(sqrt(2))` and `AB =5`
`rArr /_AOB = (pi)/(2) rArr /_OAB = 45^(@)`
Again `PA = 4, PB = 3` and `AB = 5 rArr /_BPA = 90^(@)`
Let `/_PAB = theta`.
Then `OP^(2) = OA^(2) +AP^(2) -2. OA. AP cos (45^(@) +-theta)`
`= (25)/(2) +16 -2 (5)/(sqrt(2)) 4(1)/(sqrt(2)) (cos theta +- sin theta)`
`= (57)/(2) -20 ((4)/(5)+-(3)/(5)) = (49)/(2)` or `(1)/(2)`