Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
x2 + 2y2 - 2x + 12y + 10 = 0
Given that we need to find the centre, lengths of axes, eccentricity and foci of the ellipse x2 + 2y2 - 2x + 12y + 10 = 0.
⇒ x2 + 2y2 - 2x + 12y + 10 = 0
⇒ (x2 - 2x + 1) + 2(y2 + 6y + 9) - 9 = 0
⇒ (x - 1)2 + 2(y + 3)2 = 9
⇒
⇒
Comparing with the standard form
⇒ Centre = (p,q) = (1, - 3)
Here a2>b2
⇒ eccentricity(e) =
⇒
⇒
⇒
⇒
Length of the major axis 2a = 2(3) = 6
Length of the minor axis 2b = = 3
⇒ Foci = (p±ae,q)
⇒ Foci =
⇒ Foci =