Question

Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:

x² + 4y² - 4x + 24y + 31 = 0

Answer 1

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 = 2\(\left(\frac32\right)\) = 3

⇒ Foci = (p ± ae, q)