Find the points on the ellipse 4x^2 + y^2 = 4 that are farthest away from the point (−1, 0).

zetHeernetonsgeu

zetHeernetonsgeu

Answered question

2022-12-17

Find the points on the ellipse 4 x 2 + y 2 = 4 that are farthest away from the point (−1, 0).

Answer & Explanation

Bruno Owens

Bruno Owens

Beginner2022-12-18Added 6 answers

Given:
4 x 2 + y 2 = 4
Solution:
y 2 = 4 4 x 2 y = 4 4 x 2 y = ± 2 1 x 2
The distance d(x) between (x,y) and (-1,0) can be expressed as
d ( x ) = ( x + 1 ) 2 + y 2 = ( x + 1 ) 2 + 4 4 4 x 2 = x 2 + 2 x + 1 + 4 4 x 2 = 3 x 2 + 2 x + 5
Let us maximize f ( x ) = 3 x 2 + 2 x + 5
f ( x ) = 6 x + 2 = 0 6 x = 2 x = 1 3 f ( x ) = 6
Since y = ± 2 1 ( 1 3 ) 2 = ± 2 1 1 9 = ± 2 8 9 = ± 2 2 2 3 = ± 4 2 3
The farthest point ( 1 3 , ± 4 2 3 )

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