Problem6:Find coefficients a,b,c, and d so that the curve, 
> ;
2 2 a x + a y + b x + c y + d = 0
goes through the points: (4,5), (2,7) and (4,3)
Plot this curve.

Each point (x,y) produces an equation on the variables (a,b,c,d) with, 
> Eq := (x,y) > (x^2+y^2)*a + x*b + c*y + d = 0;
2 2 Eq := (x, y) > (x + y ) a + x b + c y + d = 0
Applying this function to the given points we get, 
> Eq(4,5); Eq(2,7); Eq(4,3);
41 a  4 b + 5 c + d = 0 53 a  2 b + 7 c + d = 0 25 a + 4 b  3 c + d = 0
we can solve for a,b,c and d with, 
> solve({Eq(4,5),Eq(2,7),Eq(4,3)},{a,b,c,d});
{b = 2 a, c = 4 a, a = a, d = 29 a}
Thus, by substituting b,c and d in terms of a we obtain the curve, 
> C := subs({b = 2*a, c = 4*a, d = 29*a},Eq(x,y));
2 2 C := (x + y ) a  2 x a  4 a y  29 a = 0
so for any a not 0 we can divide through by a to obtain, 
> C := simplify(subs(a=1,C));
2 2 C := x + y  2 x  4 y  29 = 0
this is the equation of a circle on the xy plane of radius R centered at the point (x0,y0)... 
> Cir := expand( (xx0)^2 + (yy0)^2  R^2 = 0);
2 2 2 2 2 Cir := x  2 x x0 + x0 + y  2 y y0 + y0  R = 0
and we can see (comparing C with Cir) that, (x0,y0) = (1,2) and the radius is, 
> solve(R^25=29,R);
1/2 1/2 34 , 34
since R > 0 we have that the curve is a circle on the xy plane centered at (1,2) of radius sqrt(34). Can you plot it? 