Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
3x2 + 4y2 - 12x - 8y + 4 = 0
Given that we need to find the centre, lengths of axes, eccentricity and foci of the ellipse 3x2 + 4y2 - 12x - 8y + 4 = 0.
⇒ 3x2 + 4y2 - 12x - 8y + 4 = 0
⇒ 3(x2 - 4x + 4) + 4(y2 - 2y + 1) - 12 = 0
⇒ 3(x - 2)2 + 4(y - 1)2 = 12
⇒
⇒
Comparing with the standard form
⇒ Centre = (p,q) = (2,1)
Here a2>b2
⇒ eccentricity(e) =
⇒
⇒
⇒
Length of the major axis 2a = 2(2) = 4
Length of the minor axis 2b = = 2
⇒ Foci = (p±ae,q)
⇒ Foci =
⇒ Foci =
⇒ Foci = (3,1) and (1,1)