Find the equation of the ellipse having its centre at the point (2, –3), one focus at (3, –3) and one vertex at (4, –3).
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3x 2 + 4y 2 – 12x + 24y + 36 = 0
Sol. C ≡ (2, –3), S ≡ (3, –3) and A ≡ (4, –3)
Now CA = 
= 2 ∴ a = 2
Again CS = 
= 1
∴ ae = 1 ; ∴ e = 
Let the directrix cut the major-axis at Q. Then 
= e = 
If Q ≡ ( α, β ), then SA : AQ = e : 1 = 1 : 2
∴ A ≡ 
= (4, –3) ⇒ 
= 4 ; 
= –3
∴ α = 6, β = –3
Slope of CA = 0, therefore directrix will be parallel to y-axis.
Since directrix is parallel to y-axis and it passes through Q(6, –3)
∴ equation of the directrix is x = 6
Let P(x, y) be any point on the ellipse, then
e = 
= 
or 3x 2 + 4y 2 – 12x + 24y + 36 = 0
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