MathematicsIf 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  (0,2)   and Q is any point on the circle  x 2 + y 2 5xy+5=0  , then what is the maximum value of  (PQ) 2  ?


  1. A
    12 + 3
  2. B
    12 + 33
  3. C
    14 + 33
  4. D
    14 + 53 

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    Solution:

    To find the centre and radius of the circle, we have
    Comparing the above equation with general equation ( x 2 a 2 )+( y 2 b 2 )= r 2  , we have the centre as 5 2 , 1 2  , and the radius as 3 2  .
    Now, the coordinates of Q and P are (a+rcosQ,b+rsinQ)= 5 2 + 3 2 cosQ, 1 2 + 3 2 sinQ   and (0,2)  .
    By using the distance formula, the distance between P and Q is given by,
    For the maximum value of (PQ) 2  , the values of sines and cosines must be equal. So,
    cosQ=sinQ 1= sinQ cosQ tanQ=1 tanQ=tan 45 ° Q= 45 °  
    Therefore, we have cos 45 ° =sin 45 ° = 1 2  . Then,
    (PQ) 2 = 25 2 + 3 2 +5 3 2 1 2 + 1 2 = 25 2 + 3 2 +5 3 2 2 2 = 28 2 +5 3 2 ( 2 ) =14+5 3  
    Hence, the correct option is 4.
     
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