Find a rational parametrization of an affine conic section

Question

Question: construct a rational parametrization of an affine conic $$-12x^2 - 44xy -65y^2+10y-1=0.$$

My ideas: say $y = t(x+1)$ and substitute into equation.

Answer 1

First I find a point in M2

needsPackage "RationalPoints2"
QQ[x,y,z]; I = homogenize(ideal(-12*x^2 - 44*x*y -65*y^2+10*y-1), z);
rationalPoints(variety I, Bound=>12) -- ProjectivePoint{1, -1/2, -5/2}

or $(-2/5:1/5:1)$ or $(-2/5,1/5)$

Then in maxima CAS I move the point to the origin

ratexpand((-65*(y+1/5)^2)-44*(x-2/5)*(y+1/5)+10*(y+1/5)-12*(x-2/5)^2-1);

Then I set $y=tx$

solve((-65*(t*x)^2)-44*x*(t*x)+(8*(t*x))/5-12*x^2+(4*x)/5,x); # [x = (8*t+4)/(325*t^2+220*t+60),x = 0]

Then I translate back $$(x,y)=(\frac{8t+4}{325t^2+220t+60}-\frac25,t\frac{8t+4}{325t^2+220t+60}+\frac15).$$