Finding Parameters of Axis Aligned Parabola from Explicit Coordinate Equation

1. As given in the Derivation of Standard and Explicit Coordinate Equations for Axis Aligned Parabolas, the Explicit Coordinate Equation for Parabolas having their Directrix Parallel to \(X\)-Axis is given as
   
   \(y= Ax^2+Bx+C\)   ...(1)
   
   In equation (1) Variable \(y\) has only the Linear Term. This Variable will be refered to as Variable with Only Linear Term (VLT) below.
   
   The equation (1) can be converted to Standard Coordinate Equation of Axis Aligned Parabola by Completing the Squares as given in the following
   
   \(y= Ax^2+Bx+C\)
   
   \(\Rightarrow \frac{y}{A}= x^2+\frac{B}{A}x+\frac{C}{A}\)
   
   \(\Rightarrow \frac{y}{A}= x^2+\frac{B}{A}x + {(\frac{B}{2A})}^2 - {(\frac{B}{2A})}^2 +\frac{C}{A}\)
   
   \(\Rightarrow \frac{y}{A}= {(x+\frac{B}{2A})}^2 + \frac{C}{A}- {\frac{B^2}{4A^2}}\)
   
   \(\Rightarrow \frac{y}{A}= {(x+\frac{B}{2A})}^2 + \frac{4AC-B^2}{4A^2}\)
   
   \(\Rightarrow \frac{y}{A}-\frac{4AC-B^2}{4A^2}= {(x+\frac{B}{2A})}^2 \)
   
   \(\Rightarrow {(x+\frac{B}{2A})}^2=\frac{1}{A}(y-\frac{4AC-B^2}{4A})\)   ...(2)
   
   The equation (2) given above is same as the Standard Coordinate Equation of Axis Aligned Parabolas having their Directrix Parallel to \(X\)-Axis given as follows
   
   \({(x-x_v)}^2 = 4f(y - y_v)\)   ...(3)
   
   where,
   
   \(x_v=-\frac{B}{2A}\)   ...(4)
   
   \(y_v=\frac{4AC-B^2}{4A}\)   ...(5)
   
   \(4f=\frac{1}{A}\hspace{.5cm}\Rightarrow f=\frac{1}{4A}\)   ...(6)
   
   Using equations (2), (3), (4), (5) and (6) the Various Parameters of the Parabolas are given as
   1. Coordinates of the Vertex: \((x_v,y_v)\)=\((-\frac{B}{2A},\frac{4AC-B^2}{4A})\)
   2. Signed Focal Length \(f\) : \(\frac{(Co-efficient\hspace{.2cm}of\hspace{.2cm}VLT)}{4}\)=\(\frac{1}{4A}\)
   3. Length for Latus Rectum : |Co-efficient of VLT|=\(|4f|\)=\(|\frac{1}{A}|\)
   4. Coordinate of the Focus: \((x_v,y_v+f)\)=\((-\frac{B}{2A},\frac{4AC-B^2}{4A}+\frac{1}{4A})\)=\((-\frac{B}{2A},\frac{4AC-B^2+1}{4A})\)
   5. Coordinates of Points of Intersection of Latus Rectum and Parabola: \((x_v-2f,y_v+f)\),\((x_v+2f,y_v+f)\)=\((-\frac{B-1}{2A},\frac{4AC-B^2+1}{4A})\),\((-\frac{B+1}{2A},\frac{4AC-B^2+1}{4A})\)
   6. Equation of Directrix: \(y=y_v-f\hspace{.3cm}\Rightarrow y=\frac{4AC-B^2}{4A}-\frac{1}{4A}\hspace{.3cm}\Rightarrow y=\frac{4AC-B^2-1}{4A}\)
   7. Equation of Base: \(y=y_v \hspace{.3cm}\Rightarrow y=\frac{4AC-B^2}{4A}\)
   8. Equation of Latus Rectum: \(y=y_v+f\hspace{.3cm}\Rightarrow y=\frac{4AC-B^2}{4A}+\frac{1}{4A}\hspace{.3cm}\Rightarrow y=\frac{4AC-B^2+1}{4A}\)
   9. Equation of Axis of Symmetry: \(x=x_v\hspace{.3cm}\Rightarrow x=\frac{-B}{2A}\)
2. Similarly, as given in the Derivation of Standard and Explicit Coordinate Equations for Axis Aligned Parabolas, the Explicit Coordinate Equation for Parabolas having their Directrix Parallel to \(Y\)-Axis is given as
   
   \(x= Ay^2+By+C\)   ...(7)
   
   In equation (7) Variable \(x\) has only the Linear Term. This Variable will be refered to as Variable with Only Linear Term (VLT) below.
   
   The equation (7) can be converted to Standard Coordinate Equation of Axis Aligned Parabola by Completing the Squares as given in the following
   
   \(x= Ay^2+By+C\)
   
   \(\Rightarrow \frac{x}{A}= y^2+\frac{B}{A}y+\frac{C}{A}\)
   
   \(\Rightarrow \frac{x}{A}= y^2+\frac{B}{A}y + {(\frac{B}{2A})}^2 - {(\frac{B}{2A})}^2 +\frac{C}{A}\)
   
   \(\Rightarrow \frac{x}{A}= {(y+\frac{B}{2A})}^2 + \frac{C}{A}- {\frac{B^2}{4A^2}}\)
   
   \(\Rightarrow \frac{x}{A}= {(y+\frac{B}{2A})}^2 + \frac{4AC-B^2}{4A^2}\)
   
   \(\Rightarrow \frac{x}{A}-\frac{4AC-B^2}{4A^2}= {(y+\frac{B}{2A})}^2 \)
   
   \(\Rightarrow {(y+\frac{B}{2A})}^2=\frac{1}{A}(x-\frac{4AC-B^2}{4A})\)   ...(8)
   
   The equation (8) given above is same as the Standard Coordinate Equation of Axis Aligned Parabolas having their Directrix Parallel to \(Y\)-Axis given as follows
   
   \({(y-y_v)}^2 = 4f(x - x_v)\)   ...(9)
   
   where,
   
   \(x_v=\frac{4AC-B^2}{4A}\)   ...(10)
   
   \(y_v=-\frac{B}{2A}\)   ...(11)
   
   \(4f=\frac{1}{A}\hspace{.5cm}\Rightarrow f=\frac{1}{4A}\)   ...(12)
   
   Using equations (8), (9), (10), (11) and (12) the Various Parameters of the Parabolas are given as
   1. Coordinates of the Vertex: \((x_v,y_v)\)=\((\frac{4AC-B^2}{4A},-\frac{B}{2A})\)
   2. Signed Focal Length \(f\) : \(\frac{(Co-efficient\hspace{.2cm}of\hspace{.2cm}VLT)}{4}\)=\(\frac{1}{4A}\)
   3. Length for Latus Rectum : |Co-efficient of VLT|=\(|4f|\)=\(|\frac{1}{A}|\)
   4. Coordinate of the Focus: \((x_v+f,y_v)\)=\((\frac{4AC-B^2}{4A}+\frac{1}{4A},-\frac{B}{2A})\)=\((\frac{4AC-B^2+1}{4A},-\frac{B}{2A})\)
   5. Coordinates of Points of Intersection of Latus Rectum and Parabola: \((x_v+f,y_v-2f)\),\((x_v+f,y_v+2f)\)=\((\frac{4AC-B^2+1}{4A},-\frac{B-1}{2A})\),\((\frac{4AC-B^2+1}{4A},-\frac{B+1}{2A})\)
   6. Equation of Directrix: \(x=x_v-f\hspace{.3cm}\Rightarrow x=\frac{4AC-B^2}{4A}-\frac{1}{4A}\hspace{.3cm}\Rightarrow x=\frac{4AC-B^2-1}{4A}\)
   7. Equation of Base: \(x=x_v \hspace{.3cm}\Rightarrow x=\frac{4AC-B^2}{4A}\)
   8. Equation of Latus Rectum: \(x=x_v+f\hspace{.3cm}\Rightarrow x=\frac{4AC-B^2}{4A}+\frac{1}{4A}\hspace{.3cm}\Rightarrow x=\frac{4AC-B^2+1}{4A}\)
   9. Equation of Axis of Symmetry: \(y=y_v\hspace{.3cm}\Rightarrow y=\frac{-B}{2A}\)