Edward, the first step is to group the x and y terms in which the equation can be rewritten as:
3x^2-42x+y^2+4y=-142 Next complete the square for the x and y variables.
3(x^2-14x)+(y^2+4y)=-142
3(x^2-14x+49)+(y^2+4y+4)-3*49-4=-142
3(x-7)^2+(y+2)^2-147-4=-142
3(x-7)^2+(y+2)^2-151=-142
+151 +151
3(x-7)^2+(y+2)=9 Divide both sides by 9
(x-7)^2/3+(y+2)^2/9=1
Hence this conic is a vertical ellipse with center at (7,-2). b=√3 and a=√9=3. Therefore, the points of the major axis are (7,1) and (7,-5); the points of the minor axis are (7+√3,-2) and (7-√3,-2)
Hope this helps.
