UPSC NDA I 2025 Mathematics Paper

Given any point on the ellipse is3sin‌α,5‌cos‌α. In the standard parametric form of an ellipse,
x=asin‌α,‌‌y=b‌cos‌α
we identify
a=3,‌‌b=5
Since (b>a), the semi-major axis is (b=5) and the semi-minor axis is (a=3). The eccentricity e of an ellipse is
e=√1−‌

(‌ semi-minor ‌)2

(‌ semi-major ‌)2

=√1−‌

a2

b2

Substitute(a=3)and(b=5):
e=√1−‌

32

52

=√1−‌

9

25

=√‌

16

25

=‌

4

5