Given that we need to find the centre, lengths of axes, eccentricity and foci of the ellipse x² + 4y² - 4x + 24y + 31 = 0.

⇒ x² + 4y² - 4x + 24y + 31 = 0

⇒ (x² - 4x + 4) + 4(y² + 6y + 9) - 9 = 0

⇒ (x - 2)² + 4(y + 3)² = 9

⇒

⇒

Comparing with the standard form

⇒ Centre = (p,q) = (2, - 3)

Here a²>b²

⇒ eccentricity(e) =

⇒

⇒

⇒

⇒

Length of the major axis 2a = 2(3) = 6

Length of the minor axis 2b = = 3

⇒ Foci = (p±ae,q)

⇒ Foci =

⇒ Foci =