Question 2049
 These are basic questions about conic and not hard at all.
 Also,you should know how to type the square or power.

 1.What is the vertex ot the parabola with the equation (y - 1)^2 = 4x ?
   Sol:  The equations as (y - k)^2 = c(x -h) or c(y - k) = (x -h)^2
         are parabolas with vertex (h,k).
         For given (y - 1)^2 = 4x = 4(x-0) , the vertex is (0,1)

2.What is the focus of the parabola with the equation (x - 1)^2 + 32= 8y? 
  Sol: As in (1), (x - 1)^2 + 32= 8y can be converted to
       (x - 1)^2 = 8(y -4), so its vertex is (1,4).

3.Find the center of a circle with the equation x^2 + y^2 + 2x + 4y - 9 = 0. 
  Sol: Make complete square: the equation converts to
       x^2 + 2x + 1 + y^2  + 4y + 4 = 9+1+4 = 14, or
       (x+1)^2 + (y+2)^2 = 14. So, the center of the circle is (-1,-2)
       and the radius is sqrt(14).
4. Find the radius of a circle with the equation x2 + y2 - 6x - 12y + 36 = 0. 
  Sol: As in (3):x^2 - 6x + (6/2)^2 + y2  - 12y + (12/2)^2 = -36 + 9 + 36 =9.
       Or (x -3)^2 + (y-6)^2 = 9.
       So, the center of the circle is (3,6)
       and the radius is 3.
5.Find the center of an ellipse with the equation 9x^2 +16y^2 -18x+64y= 71. 
Sol: Make complete square: the equation converts to
       9(x^2 - 2x + 1) + 16(y^2  + 4y + 4) = 71 + 9 + 64 = 144, or
       9(x-1)^2 + 16(y + 2)^2 = 144.  Dividing each term by 144, we have:
       (x-1)^2/16 + (y + 2)^2/9 = 1. 
     The center of the ellipse is (1,-2). 
    [also, the length of the major axis = 4 and the length of the minor axis
     = 3]

6.Find the foci of an ellipse with the equation 7(x - 2)^2 + 3(y - 2)^2 = 21. 
  Sol: Dividing each term by 21, we have:
       (x-1)^2/3 + (y - 2)^2/7 = 1. 
We see the center is (1,2) and the length of the major axis a = sqrt(7),
the length of the minor axis b = sqrt(3). Note, the  long axis is along 
y-direction(vertical).
Since c^2 = a^2-b^2 = 7-3 =4, we get c = 2.
 Hence, the foci are (1,2+c) = (1,4) and (1,2-c) = (1,0)

7.Find the length of the major and minor axes of an ellipse with the equation  Sol: As in (5), the equation converts to
       16(x^2 + 2x + 1) + 25(y^2  - 6y + 9) = 159 +16 +225 = 400 or
       4(x+1)^2 + 25(y-3)^2 = 200. 
       Dividing each term by 200, we have:
       (x+1)^2/50 + (y - 2)^2/8 = 1. 
       Hence, the major axis a = sqrtt(50) = 5 sqrt(2),
        the minor axis b = sqrtt(8) = 2 sqrt(2).


 Kenny