If a point P has coordinates  (0,−2)   and Q is any point on the circle  x 2 + y 2 −5x−y+5=0  , then what is the maximum value of  (PQ) 2  ?

# If a point P has coordinates    and Q is any point on the circle   , then what is the maximum value of   ?

1. A
12 + $\sqrt{3}$
2. B
12 + $3\sqrt{3}$
3. C
14 + $3\sqrt{3}$
4. D
14 + 5$\sqrt{3}$

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### Solution:

To find the centre and radius of the circle, we have
Comparing the above equation with general equation  , we have the centre as  , and the radius as  .
Now, the coordinates of Q and P are   and  .
By using the distance formula, the distance between P and Q is given by,
For the maximum value of  , the values of sines and cosines must be equal. So,

Therefore, we have  . Then,

Hence, the correct option is 4.

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