In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at (0,5√(3)) then the length of its latus rectum is:

Correct option is C. 5
Step 1: Use foci and difference between principle axes
Given that centre of the ellipse is at origin and one of the foci is at (0,5√(3)) ⇒Major axis of ellipse is along y axis.
⇒Let equation of ellipse as x²/a²+y²/b²=1 , where a < b
⇒ focus= (0,be)=(0,5√(3))
⇒ be=5√(3) (1)
Also given difference between lengths of major axis minor axis is 10.
⇒ 2b-2a=10
⇒ b-a=5 (2)
⋍ e=√(1-a²/b²)
⇒ b²e²=b²-a²
⇒ (b-a)(b+a)=(be)²
⇒ 5(b+a)=75 (From eq. (1) (2))
⇒ b+a=15 (3)
Step 2: Use above equations to get the required result
Solving equations (2) (3), we get
b=10 & a=5
⋍Length of latus rectum= 2a²/b
⇒Length of latus rectum=2×25/10=5
Hence, (C) is the correct option.