1. Since 25 > 4 the standard equation of the ellipse is \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\)
⇒ a2 = 25; b2 = 4
c2 = a2 – b2 = 25 – 4 = 21
⇒ c = √21
Coordinate of foci are (0, ±√21)
Coordinate of vertex are (0, ±5)
Length of major axis = 2a = 2 × 5 = 10
Length of minor axis = 2b = 2 × 2 = 4
Eccentricity = e = \(\frac{c}{a}\) =\(\frac{\sqrt{21}}{5}\)
Length of latus rectum = \(\frac{2b^2}{a} = \frac{2 \times 4}{5} = \frac{8}{5}.\)
2. Since 16 > 9 the standard equation of the ellipse is \(\frac{x^2}{a} + \frac{y^2}{b^2} = 1\)
⇒ a2 = 16; b2 = 9
c2 = a2 – b2 = 16 – 9 = 7
⇒ c = √7
Coordinate of foci are (±√7, 0)
Coordinate of vertex are (±4, 0)
Length of major axis = 2a = 2 × 4 = 8
Length of minor axis = 2b = 2 × 3 = 6
Eccentricity = e = \(\frac{c}{a}\)=\(\frac{\sqrt{7}}{4}\)
Length of latus rectum = \(\frac{2b^2}{a} = \frac{2 \times 9}{4} = \frac{9}{2}.\)