The equation of the ellipse with focus (−1, 1), directrix x − y + 3 = 0 and eccentricity 1/2 is
Solution
\[7 x^2 + 7 y^2 + 2xy - 10y + 10x + 7 = 0\]
\[\text{ Let }P(x,y)\text{ be any point on the ellipse whose focus andeccentricity are }S\left( - 1, 1 \right)\text{ and }e=\frac{1}{2},\text{ respectively. }\]
Let PM be the perpendicular from P on the directrix.
\[\text{ Then SP }= e \times \text{ PM}\]
\[ \Rightarrow SP = \frac{1}{2} \times PM\]
\[ \Rightarrow 2SP = PM\]
\[ \Rightarrow 4 \left( SP \right)^2 = P M^2 \]
\[ \Rightarrow 4\left[ \left( x + 1 \right)^2 + \left( y - 1 \right)^2 \right] = \left( \frac{x - y + 3}{\sqrt{1^2 + \left( - 1 \right)^2}} \right)^2 \]
\[ \Rightarrow 4\left[ x^2 + 1 + 2x + y^2 + 1 - 2y \right] = \frac{x^2 + y^2 + 9 - 2xy - 6y + 6x}{2}\]
\[ \Rightarrow 8 x^2 + 8 + 16x + 8 y^2 + 8 - 16y = x^2 + y^2 + 9 - 2xy - 6y + 6x\]
\[ \therefore 7 x^2 + 7 y^2 + 2xy - 10y + 10x + 7 = 0\]
This is the required equation of the ellipse.
