Question

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

4x² + 16y² - 24x - 32y - 12 = 0

Answer 1

Given that we need to find the centre, lengths of axes, eccentricity and foci of the ellipse

4x² + 16y² - 24x - 32y - 120 = 0.

⇒ 4x² + 16y² - 24x - 32y - 120 = 0

⇒ 4(x² - 6x + 9) + 16(y² - 2y + 1) - 172 = 0

⇒ 4(x - 3)² + 16(y - 1)² = 172

Comparing with the standard form

⇒ Centre = (p, q) = (3,1)

Here a² > b²

⇒ eccentricity(e) = 

Length of the major axis 2a = 2\((\sqrt{43})\) = 2\(\sqrt{43}\)

Length of the minor axis 2b = 2\(\left(\cfrac{\sqrt{43}}2\right)=\sqrt{43}\)

⇒ Foci = (p ± ae, q)