Derivation and Properties of Implicit Coordinate Equation for Axis Aligned and Arbitrarily Rotated Parabolas

1. The following gives the derivation for Implicit Coordinate Equation for Axis Aligned and Arbitrarily Rotated Parabolas.
   Let's consider a Parabola having it's Focus at a point \((x_f,y_f)\) and having Directrix given by the equation \(A_Dx + B_Dy + C_D=0\). Now,
   
   Distance of any point (\(x,y\)) on the Parabola from the Focus = \(\sqrt{{(x-x_f)}^2 + {(y-y_f)}^2}\)
   
   Also, Distance of any point (\(x,y\)) on the Parabola from the Directrix = \(\frac{A_Dx + B_Dy + C_D}{\sqrt{{A_D}^2 + {B_D}^2}}\)
   
   Now, as per the definition of Parabola, any point (\(x,y\)) on the Parabola must be equidistant from the Directrix and the Focus. Therefore,
   
   \(\sqrt{{(x-x_f)}^2 + {(y-y_f)}^2} = \frac{A_Dx + B_Dy + C_D}{\sqrt{{A_D}^2 + {B_D}^2}}\)
   
   \(\Rightarrow {(x-x_f)}^2 + {(y-y_f)}^2 = \frac{{(A_Dx + B_Dy + C_D)}^2}{{A_D}^2 + {B_D}^2}\)
   
   \(\Rightarrow x^2 + {x_f}^2 -2x_fx + y^2 + {y_f}^2 - 2y_fy = \frac{{(A_Dx + B_Dy + C_D)}^2}{{A_D}^2 + {B_D}^2}\)
   
   \(\Rightarrow ({A_D}^2 + {B_D}^2) (x^2 + {x_f}^2 -2x_fx + y^2 + {y_f}^2 - 2y_fy) ={A_D}^2x^2 + {B_D}^2y^2 + {C_D}^2 + 2A_DB_Dxy + 2A_DC_Dx + 2B_DC_Dy \)
   
   \(\Rightarrow {A_D}^2x^2 + {A_D}^2{x_f}^2 - 2{A_D}^2x_fx + {A_D}^2y^2 + {A_D}^2{y_f}^2 - 2{A_D}^2y_fy + {B_D}^2x^2 + {B_D}^2{x_f}^2 - 2{B_D}^2x_fx + {B_D}^2y^2 + {B_D}^2{y_f}^2 - 2{B_D}^2y_fy = {A_D}^2x^2 + {B_D}^2y^2 + {C_D}^2 + 2A_DB_Dxy + 2A_DC_Dx + 2B_DC_Dy \)
   
   \(\Rightarrow {A_D}^2{x_f}^2 - 2{A_D}^2x_fx + {A_D}^2y^2 + {A_D}^2{y_f}^2 - 2{A_D}^2y_fy + {B_D}^2x^2 + {B_D}^2{x_f}^2 - 2{B_D}^2x_fx + {B_D}^2{y_f}^2 - 2{B_D}^2y_fy = {C_D}^2 + 2A_DB_Dxy + 2A_DC_Dx + 2B_DC_Dy \)
   
   \(\Rightarrow {B_D}^2x^2 - 2A_DB_Dxy + {A_D}^2y^2 - 2{A_D}^2x_fx - 2{B_D}^2x_fx - 2A_DC_Dx - 2{A_D}^2y_fy - 2{B_D}^2y_fy - 2B_DC_Dy + {A_D}^2{x_f}^2 + {B_D}^2{x_f}^2 + {A_D}^2{y_f}^2 + {B_D}^2{y_f}^2 - {C_D}^2 =0\)
   
   \(\Rightarrow {B_D}^2x^2 - 2A_DB_Dxy + {A_D}^2y^2 - 2(({A_D}^2 + {B_D}^2)x_f + A_DC_D)x - 2(({A_D}^2 + {B_D}^2)y_f + B_DC_D)y + ({A_D}^2 + {B_D}^2){x_f}^2 + ({A_D}^2 + {B_D}^2){y_f}^2 - {C_D}^2 =0\)   ...(1)
   
   Now, Setting \(\mathbf{P}={A_D}^2 + {B_D}^2\) in equation (1) we get
   
   \({B_D}^2x^2 - 2A_DB_Dxy + {A_D}^2y^2 - 2(Px_f + A_DC_D)x - 2(Py_f + B_DC_D)y + P{x_f}^2 + P{y_f}^2 - {C_D}^2 =0\)   ...(2)
   
   Now, Setting \(\mathbf{A}={B_D}^2\), \(\mathbf{B}=-2A_DB_D\), \(\mathbf{C}={A_D}^2\), \(\mathbf{D}=-2(Px_f + A_DC_D)\), \(\mathbf{E}=-2(Py_f + B_DC_D)\), \(\mathbf{F}=P{x_f}^2 + P{y_f}^2 - {C_D}^2\) in equation (2) we get
   
   \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F =0\)   ...(3)
   
   The equation (3) given above is a General Quadratic Equation in 2 Variables and gives the Implicit Coordinate Equation for Arbirarily Rotated Parabolas.
2. For Axis Aligned Parabola having Directrix Parallel to \(X\)-Axis the Coefficient of Variable \(x\) (i.e. \(A_D\)) of the Equation of Directrix is 0. Hence, the equation (1) becomes
   
   \({B_D}^2x^2 - 2{B_D}^2x_fx - 2({B_D}^2y_f + B_DC_D)y + {B_D}^2{x_f}^2 + {B_D}^2{y_f}^2 - {C_D}^2 =0\)   ...(4)
   
   Now, Setting \(\mathbf{A}={B_D}^2\), \(\mathbf{B}=-2{B_D}^2x_f\), \(\mathbf{C}=-2({B_D}^2y_f + B_DC_D)\), \(\mathbf{D}={B_D}^2{x_f}^2 + {B_D}^2{y_f}^2 - {C_D}^2\) in equation (4) we get
   
   \(Ax^2 + Bx + Cy + D =0\)   ...(5)
   
   Equation (5) above gives the Implicit Coordinate Equation of Parabolas having Directrix Parallel to \(X\)-Axis.
3. For Axis Aligned Parabola having Directrix Parallel to \(Y\)-Axis the Coefficient of Variable \(y\) (i.e. \(B_D\)) of the Equation of Directrix is 0. Hence, the equation (1) becomes
   
   \({A_D}^2y^2 - 2({A_D}^2x_f + A_DC_D)x - 2{A_D}^2y_fy + {A_D}^2{x_f}^2 + {A_D}^2{y_f}^2 - {C_D}^2 =0\)   ...(6)
   
   Now, Setting \(\mathbf{A}={A_D}^2\), \(\mathbf{B}=-2({A_D}^2x_f + A_DC_D)\), \(\mathbf{C}=-2{A_D}^2y_f\), \(\mathbf{D}={A_D}^2{x_f}^2 + {A_D}^2{y_f}^2 - {C_D}^2\) in equation (6) we get
   
   \(Ay^2 + By + Cx + D =0\)   ...(7)
   
   Equations (7) above gives the Implicit Coordinate Equation of Parabolas having Directrix Parallel to \(Y\)-Axis.
4. Following are some Properties of Implicit Coordinate Equation of Parabolas
   1. The Square Term must be Present for atleast One Variable.
   2. If \(xy\) Term is Present, then the Directrix of the Parabola is Not Parallel Any Coordinate Axes (i.e. the Parabola is Rotated). The \(xy\) Term is Present Only When Square Terms of Both Variables are Present. Conversely, the \(xy\) Term will Always be Present if Square Terms of Both Variables are Present.
   3. If Square Terms of Both Variables are Present then they Always have the Same Sign i.e. either Both are Positive or Both are Negative
   4. If \(xy\) Term is Not Present, then the Directrix of the Parabola is Parallel to one of the Coordinate Axes (i.e. the Parabola is Axis Aligned). Please note that For Axis Aligned Parabolas Square Term is Present for Only 1 Variable and Only Linear Term Must be Present for the Other Variable. For such Parabolas the Directrix is Parallel to the Coordinate Axis corresponding to the Variable having the Square Term and the Parabola is Symmetric to a Line Parallel to the Coordinate Axis Corresponding to the Variable having Only the Linear Term.