Determine the parametr [i]a[/i] so that the ellipse [math]4x^2-4xy+5y^2=16[/math] is transformed to the circle.

[b]Experimental method:[/b][br]1. Construct the slider [icon]/images/ggb/toolbar/mode_slider.png[/icon] for unknown parameter [i]a[/i].[br]Place the 2 x2 matrix of linear trasformations [code]A = {{1, a}, {0, 1}}[/code][br]2. Construct the image of the conic: Command [code]ApplyMatrix(A,c)[/code] where [code]c[/code] is the conic[br]3. Change the value of slider a to get the circle as a image of the ellipse.[br]4. [i]a[/i] = -1/2[br][br][b]Algebraic method[br][/b]Tranformation equations x'=x+ay; y'=y yield inverse mapping x=x'-ay'; y=y'.[br][math]4x^2-4xy+5y^2=16[/math].[br]Substitute x and y in the implicit equation of the ellipse.[br][math]4\left(x'-ay'\right)^2-4y'\left(x'-ay'\right)+5y'^2=16[/math], rewritte the equation in more convenient to determine the quadratic coefficients:[br][math]4x'^2+y'^2\left(4a^2+4a+1\right)-4x'y'\left(2a+1\right)=16[/math][br]Circle has same axis and zero mixed member x'y'. Equations [math]4=\left(4a^2+4a+1\right);\left(2a+1\right)=0[/math] yield one solution a=-1/2.[br]