General equation of ellipse:
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Let equation of directrix of an ellipse be
ax + by + c = 0 and the focus be (h, k).
Let the eccentricity of ellipse be e(e < 1).
If P(x, y) is any point on ellipse, then
PS2 = e2 PM2
=> (x - h)2 + (y - k)2 = e2 
=> which is of the form
=> ax2+2hxy +by2+2gx+2fy+c= 0, ... (*) where
D = abc +2 fgh -af2-bg2 - ch2 ≠ 0, h2 < ab ,
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Which are the essential and sufficient condition for a general quadratic equation which is given by (*) to represent an ellipse .
Example: Find eccentricity and latus rectum of ellipse 4x2 + 9y2 - 8x - 36y + 4 = 0, also find centre, focus and directrix.
Solution: On simplification, the equation becomes
4(x - 1)2 + 9(y - 2)2 = 36
Then, 
, here X = x - 1 and Y = y - 2, a = 3, b = 2
Now, e = 
The latus rectum = 
For a > b
Focus (X, Y) is (ae, 0) and (-ae, 0)
=> (x, y) for focus Þ (1 + ae, 2) and (1 - ae, 2)
=> foci are (1 + √5, 2) and (1 - √5, 2)
Likewise equation of directrices are 
are directrices.
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