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Q4 c1 is the circle x² + y² -4y-20=0 and l is the line 2x-y+5=0

Q4 c1 is the circle x² + y² -4y-20=0 and l is the line 2x-y+5=0

(a) write down the centre and radius lenght of c1

(b) Find the perpendicular distance from the centre of c1 to the line l in surd form

(ii) Hence or otherwise find PQ in surd form

(c) PQ is the diameter of circle c2
     Find the equation of c2
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1 Answer

 X^2 + Y^2 - 4y -20 =0
 
   X^2 + Y^2 - 4Y +4 = 24
 
   X^2 + ( Y - 2 ) ^2 = 24
 
  c1 : ( 0 , 2)
 
   b)   L1 : 2X - y + 5 = 0
 
       L2  :   Y = -1/2 X +b
 
                2 = -1/2 (0) +b
    
          L2 :  Y = -1/2 X + 2  
 
 
           L1 :     2X - Y = -5
           L2       1/2X+ Y = 2
 
                      5/ 2 X = -3
 
                      X = - 6/5      Y = 13/5
 
                      PQ :   ( 0, 2 ) ( -6/5 , 13/5)
 
                              d =√( ( -6/5 -0) ^2 ) + ( 13/5 - 2 ) ^2 )=
 
                                  =  √ ( 36/ 25) + ( 9/ 25) = √(45)/ 5 = 3√5/ 5 
 
                c ) :
 
                    ( 0, 2 )   ( -6/5 , 13/5)
 
                     m = ( 13/5  - 2 ) / ( -6/5 - 0) = - 1/2
 
                       Y = -1/2 X + b
 
                       2 = -1/2(0) + b   , b= 2
 
                      Y = - 1/2 X + 2