Let e be the eccentricity of the ellipse having focus F(p, q) and equation of directrix ax + by + c = 0
Let P(x, y) be any point on the ellipse and PM be the perpendicular on the directrix.
Therefore,
PF/PM = e
PF2 = e2 x (PM)2
(x – p)2 + (y - q)2 = e2. (ax + by + c)2
(a2 + b2)
Simplifying we get;
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
Where, h2 < ab
And.
abc + 2gh – af2 – bg2 – ch2 = 0
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