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

Question

Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse: 4x 2 + y 2 - 8x + 2y + 1 = 0

Philoid · Accepted Answer

Given that we need to find the centre, lengths of axes, eccentricity and foci of the ellipse 4x² + y² - 8x + 2y + 1 = 0.

⇒ 4x² + y² - 8x + 2y + 1 = 0

⇒ 4(x² - 2x + 1) + (y² + 2y + 1) - 4 = 0

⇒ 4(x - 1)² + (y + 1)² = 4

⇒

Comparing with the standard form

⇒ Centre = (p,q) = (1, - 1)

Here b²>a²

⇒ eccentricity(e) =

⇒

Length of the major axis 2b = 2(2) = 4

Length of the minor axis 2a = 2(1) = 2

⇒ Foci = (p,q±be)

⇒ Foci =