Equation of the Ellipse

To find the equation of an ellipse, let P\left( {x,y} \right) be any point of the ellipse and M\left(  {\frac{a}{e},y} \right) the corresponding point on the directrix as shown in the given diagram, then by definition of ellipse, we have


equation-ellipse

\begin{gathered} \frac{{PF}}{{PM}} = e \\ \Rightarrow PF = ePM \\ \Rightarrow \sqrt {{{\left( {x - ae}  \right)}^2} + {{\left( {y - 0} \right)}^2}}   = e\sqrt {{{\left( {x - \frac{a}{e}} \right)}^2} + {{\left( {y - y} \right)}^2}} \\ \Rightarrow {\left( {x - ae} \right)^2} +  {y^2} = {e^2}{\left( {x - \frac{a}{e}} \right)^2} \\ \Rightarrow {x^2} - 2aex + {a^2}{e^2} +  {y^2} = {e^2}\left( {{x^2} - 2\frac{a}{e}x + \frac{{{a^2}}}{{{e^2}}}} \right) \\ \Rightarrow {x^2} - 2aex + {a^2}{e^2} +  {y^2} = {e^2}{x^2} - 2aex + {a^2} \\ \Rightarrow {x^2} - {e^2}{x^2} + {y^2} =  {a^2} - {a^2}{e^2} \\ \Rightarrow \left( {1 - {e^2}} \right){x^2}  + {y^2} = {a^2}\left( {1 - {e^2}} \right) \\ \Rightarrow \frac{{{x^2}}}{{{a^2}}} +  \frac{{{y^2}}}{{{a^2}\left( {1 - {e^2}} \right)}} = 1\,\,\,\,{\text{ -  -  -  }}\left( {\text{i}} \right) \\ \end{gathered}


It is clear from the given diagram that in the triangle, FOB, we have the relation given as

\begin{gathered} {a^2}{e^2} + {b^2} = {a^2} \\ \Rightarrow {a^2} - {a^2}{e^2} = {b^2} \\ \Rightarrow {a^2}\left( {1 - {e^2}} \right)  = {b^2} \\ \end{gathered}


Using this relation in equation (i), we have

\frac{{{x^2}}}{{{a^2}}}  + \frac{{{y^2}}}{{{b^2}}} = 1


This is the equation of the ellipse whose centre is at origin and foci lie on the X-axis. The lengths of semi-major and semi-minor axes of this ellipse are a and b respectively.
If foci lie on the Y-axis, then its graph as shown in the given diagram. In this case the equation of ellipse will be

\frac{{{x^2}}}{{{b^2}}}  + \frac{{{y^2}}}{{{a^2}}} = 1

NOTE: For we use the relation {a^2} - {b^2} = {a^2}{e^2}

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