Find the equation of the ellipse in the case:
focus is (0, 1), directrix is x + y = 0 and e = \[\frac{1}{2}\] .
Solution
\[\text{ Let S(0, 1) be the focus and ZZ' be the directrix . } \]
\[\text{ Let P(x, y) be any point on the ellipse and let PM be the perpendicular from P on the directrix } . \]
\[\text{ Then by the definition, we have: } \]
\[SP = e \times PM\]
\[ \Rightarrow SP = \frac{1}{2} \times PM\]
\[ \Rightarrow 2SP = PM\]
\[ \Rightarrow 4 \left( SP \right)^2 = {PM}^2 \]
\[ \Rightarrow 4\left[ \left( x \right)^2 + \left( y - 1 \right)^2 \right] = \left| \frac{x + y}{\sqrt{1^2 + \left( 1 \right)^2}} \right|^2 \]
\[ \Rightarrow 4\left[ x^2 + y^2 + 1 - 2y \right] = \frac{x^2 + y^2 + 2xy}{2}\]
\[ \Rightarrow 8 x^2 + 8 y^2 + 8 - 16y = x^2 + y^2 + 2xy\]
\[ \Rightarrow 7 x^2 + 7 y^2 - 2xy - 16y + 8 = 0\]
\[\text{ This is the required equation of the ellipse.} \]
