Question

Question 4 Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse x^2/25 + y^2/100 = 1

Class X1 - Maths -Conic Sections Page 255

Answer 1

concept : if equation of ellipse is x²/b² + y²/a² = 1 ( b < a ) then,
vertices ( 0, ± a)
foci ( 0, ± c ) where, c² = a² - b²
Length of minor axis = 2b
length of major axis = 2a
eccentricity ( e ) = c/a
length of latusrectum = 2b²/a

Here, x²/25 + y²/100 = 1 or, x²/5² + y²/10² = 1
∵ denominator of x²/25 is smaller than the denominator of y²/100 .
∴ The major axis is along Y-axis .
on comparing given equation with x²/b² + y²/a² = 1 , we get
a = 10 and b = 5
now, c² = a² - b² = 10² - 5² = 100 - 25
c² = 75 => c = 5√3

so, vertices ( 0, ± a) = ( 0, ± 10)
foci ( 0, ± c ) = ( 0, ± 5√3)
length of major axis = 2a = 2 × 10 = 20
length of minor axis = 2b = 2 × 5 = 10
eccentricity ( e ) = c/a = 5√3/10 = √3/2
length of latusrectum = 2b²/a = 2 × 25/10 = 5