Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
4x2 + 16y2 - 24x - 32y - 12 = 0
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() =
Length of the minor axis 2b =
⇒ Foci = (p±ae,q)
⇒ Foci =
⇒ Foci =