Expert: Steve Holleran - 5/21/2007

Question
hi i am faced with the following question. (i)Find the coordinates of the centre and the radius of the circle x^2+2x+y^2-4y=4  (ii)by writing x+1=3sin@, show that the parametric equations of this circle are x=-1+3sin@, y=2+3cos@; NB:(@='theta')  (iii)show that the x-coodinates of the points of intersection of this circle with the line x+y=1 are x=-1 +  (3/2) sqrt(2)
              -
ok i tried part (i)and i think it is right. i got out (using completing the square) (x+1)^2+(y-2)^2=(3)^2 centre =(-1,2), radius=3 now after this i am not sure how to proceed.i know the general equations for a circle with centre (a,b) and radius r  are x=a+rcos@, y=b+rsin@ BUT for part (ii) they ask to 'SHOW'..... so i don't think it is as easy as just plugging in the values i got in (i). thank you for your help in advance!!!

Answer
Hi Jon,

You are fine on part (i).  

For part (ii), the only thing I can see is that you are to make the substitution they suggest, and get y to come out to the value wanted:

  (x + 1)^2 + (y-2)^2 = 9

  (-1 + 3 sin@ + 1)^2 + (y - 2)^2 = 9

  (3 sin@)^2 + (y - 2)^2 = 9

   9 sin^2@  + (y - 2)^2 = 9

               (y - 2)^2 = 9 - 9 sin^2 @

                         = 9 (1 - sin^2 @)

                         = 9 cos^2 @

so               y - 2    = 3 cos @

and                     y = 2 + 3 cos @

Then for part (iii), you have :

         (x + 1)^2 + (y - 2)^2 = 9

and                x + y = 1 ----> y = 1 - x

substituting,   (x + 1)^2 + (1 - x - 2)^2 = 9

               (x + 1)^2 + (-x - 1)^2 = 9

Now factor a -1 out of the second parenthesis:

               (x + 1)^2 + (-1)^2(x + 1)^2 = 9

                      2 (x + 1)^2 = 9

                        (x + 1)^2 = 9 / 2

                   x + 1  = 3 / sqrt(2) = 3 sqrt(2)/ 2

                         x = -1 + 3/2 *  sqrt(2)


I hope this is what you needed.

Steve Holleran