General Equation of a Conic Section When Its Focus, Directrix and Eccentricity are Given
Category : JEE Main & Advanced
Let \[S(\alpha ,\beta )\] be the focus, \[Ax+By+C=0\] be the directrix and \[e\] be the eccentricity of a conic. Let \[P(h,k)\] be any point on the conic. Let PM be the perpendicular from P, on the directrix. Then by definition,
\[SP=ePM\]\[\Rightarrow \] \[S{{P}^{2}}={{e}^{2}}P{{M}^{2}}\]
\[\Rightarrow \] \[{{(h-\alpha )}^{2}}+{{(k-\beta )}^{2}}={{e}^{2}}{{\left( \frac{Ah+Bk+C}{\sqrt{{{A}^{2}}+{{B}^{2}}}} \right)}^{2}}\]
Thus the locus of \[(h,k)\] is \[{{(x-\alpha )}^{2}}+{{(y-\beta )}^{2}}=\]\[{{e}^{2}}\frac{{{(Ax+By+C)}^{2}}}{({{A}^{2}}+{{}^{2}})}\]
which is general equation of second degree.
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