Conic Section Translation

1. Conic Section Translation refers to Changing the Position of a Conic Section Object. This is done by translating the General Quadratic Equation in 2 Variables representing the Conic Section.
2. The General Quadratic Equation in 2 Variables representing a Conic Section is given as follows
   
   \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\)   ...(1)
   
   On translating the above equation (1) by an offset \((x_t,y_t)\) as per the Rule of Translation of Equations, the updated equation is given as
   
   \(A{(x-x_t)}^2 + B(x-x_t)(y-y_t) + C{(y-y_t)}^2 + D(x-x_t) + E(y-y_t) + F = 0\)
   
   \(\Rightarrow A(x^2 + {x_t}^2 - 2x_tx) + B(xy - y_tx - x_ty + x_ty_t) + C(y^2 + {y_t}^2 - 2y_ty) + Dx - Dx_t + Ey - Ey_t + F = 0\)
   
   \(\Rightarrow Ax^2 + A{x_t}^2 - 2Ax_tx + Bxy - By_tx - Bx_ty + Bx_ty_t + Cy^2 + C{y_t}^2 - 2Cy_ty + Dx - Dx_t + Ey - Ey_t + F = 0\)
   
   \(\Rightarrow Ax^2 + Bxy + Cy^2 + (D - 2Ax_t - By_t)x + (E - 2Cy_t - Bx_t)y + A{x_t}^2 + Bx_ty_t + C{y_t}^2 - Dx_t - Ey_t + F = 0\)   ...(2)
   
   \(\Rightarrow Ax^2 + Bxy + Cy^2 + D_1x + E_1y + F_1 = 0\)   ...(3)
   
   where
   
   \(D_1=D - 2Ax_t - By_t\)
   
   \(E_1=E - 2Cy_t - Bx_t\)
   
   \(F_1= A{x_t}^2 + Bx_ty_t + C{y_t}^2 - Dx_t - Ey_t + F\)
   
   The equations (2) and (3) above give the Equation of the Conic Section Translated by an Offset \((x_t,y_t)\).
   
   Please note the Translating a Conic Section Does Not Change the Values of its Quadratic Co-efficients (\(x^2, xy\) and \(y^2\)). However it Changes the Values of its its Linear Co-efficients (\(x\) and \(y\)) and the Constant of the Equation.