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

Question

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

Answer 1

 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