I. 8. Cicle as a linear image of the ellipse
Determine the parametr a so that the ellipse is transformed to the circle.
Experimental method:
1. Construct the slider 
for unknown parameter a.
Place the 2 x2 matrix of linear trasformations
for unknown parameter a.
Place the 2 x2 matrix of linear trasformations A = {{1, a}, {0, 1}}
2. Construct the image of the conic: Command ApplyMatrix(A,c) where c is the conic
3. Change the value of slider a to get the circle as a image of the ellipse.
4. a = -1/2
Algebraic method
Tranformation equations x'=x+ay; y'=y yield inverse mapping x=x'-ay'; y=y'.
.
Substitute x and y in the implicit equation of the ellipse.
, rewritte the equation in more convenient to determine the quadratic coefficients:
Circle has same axis and zero mixed member x'y'. Equations yield one solution a=-1/2.