Let P (x, y) be a point on the ellipse x ^ 2 + 4Y ^ 2 = 16, then the maximum value of X + y is equal to
x²/16+y²/4=1
So x = 4cosm
y=2sinm
Then x + y = 2sinm + 4cosm = √ (2 & # 178; + 4 & # 178;) sin (M + n)
tann=4/2
So the maximum is 2 √ 5
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