Question 1690
1. 2x^2-2y^2+20y-75=0 
   Sol: To complete the square : 2x^2-2(y^2-10y + (10/2)^2)=75-50 ,
        So, 2x^2 - 2(y-5)^2 =25
        Or x^2 - (y-5)^2 = 25/2    
        Or x^2/(25/2) - (y-5)^2/(25/2) = 1
        a) This is a hyperbola center at (0,5)
        b) Standard form : x^2/(25/2) - (y-5)^2/(25/2) = 1
        c) a hyperbola centered at (0,5), its axis is y=5.
          (I am notsure what you mean about properties.)
        d) when y =0 , x^2 = 25/2 + 25 = 75/2, so x= sqrt(75/2) or -sqrt(75/2)
           (two x-intercepts).
           when x = 0, -(y-5)^2 = 25/2 has no solutions ,so no y-intercepts.
        e) Graph of the hyperbola is left for you.(It's easy.)


2. 9x^2+4y^2-36x+8y+31=0
   Sol: By complete square : 9(x^2-4x+4) +4(y^2+2y +1) =-31+36+4 ,
        Or 9(x-2)^2 + 4(y+1)^2 = 9,
        Or (x-2)^2 + (y+1)^2/(9/4) = 1,
        a) This is an ellipse centered at (2,-1)
        b) Standard form : (x-2)^2 + (y+1)^2/(9/4) = 1,
        c) an ellipse centered at (2,-1), its long axis a=3/2 is on y=-1,
          short axis b is on x=2.
          (I am notsure what you mean about properties.)
        d)when y =0 , (x-2)^2 = 1-4/9 = 5/9, so x = 2 + sqrt(5)/3, or 
           x = 2-sqrt(5)/3. (two  x-intercepts)
          when x =0, (y+1)^2 = 9/4(1-4). No solutions, so no y-intercepts.
         e) Graph of the ellipse is left for you.

 Kenny