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Finding Equation of Ellipse from a given Directrix, Adjacent Focus and Eccentricity

  1. Given the Equation of a Directrix \(Ax + By + C=0\), Coordinates of it's Adjacent Focus \((x_{f1},y_{f1})\) and Eccentricity \(e\) the following gives the steps for calculation of the Equation of the Ellipse
  2. Calculate the Signed Distance \(d\) Between the given Focus and the Directrix as follows

    \(d=\frac{Ax_{f1} + By_{f1} + C}{\sqrt{A^2 +B^2}}\)   ...(1)
  3. Calculate the Coordinates of Projection of the Focus on the Directrix as follows

    \(\begin{bmatrix}x_p\\y_p\end{bmatrix}=\begin{bmatrix}x_{f1}\\y_{f1}\end{bmatrix} - d\begin{bmatrix}\frac{A}{\sqrt{A^2 +B^2}}\\\frac{B}{\sqrt{A^2 +B^2}}\end{bmatrix}\)   ...(2)
  4. Calculate the Coordinates of the Vertex \((x_{v1},y_{v1})\) lying between Focus and Directrix as follows

    We know that the Coordinates of Vertex \((x_{v1},y_{v1})\) Divides the Line Joining the Coordinates of Focus \((x_{f1},y_{f1})\) and Coordinates of Projection of the Focus on the Directrix \((x_p,y_p)\) intenally in a Ratio \(e:1\). Therefore using Section Formula we have

    \(\begin{bmatrix}x_{v1}\\y_{v1}\end{bmatrix}=\begin{bmatrix}\frac{x_{f1} + ex_p}{e+1}\\\frac{y_{f1} + ey_p}{e+1}\end{bmatrix}\)   ...(3)
  5. Once we get Coordinates of Both the Adjacent Focus and the Vertex the Equation of the Ellipse can be determined as given in the topic Finding Equation of Ellipse from a given Focus, a Vertex and Eccentricity.
Related Topics
Finding Equation of Ellipse from given 2 Foci and Major Axis Length,    Finding Equation of Ellipse from a given Focus, a Vertex and Eccentricity,    Finding Parametric Equations for Axis Aligned and Rotated Ellipse,    Introduction to Ellipse and Imaginary Ellipse,    General Quadratic Equations in 2 Variables and Conic Sections
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