x^2 + y^2/4 = 1 using implicit differentiation we have that
2x + (1/2)yy' = 0
y' = -2x / (1/2 y)
y' = -4x/ y
Yhe slope of the line = 2
So....we want to find where
-4x / y = 2
-4x = 2y
-2x = y
Sub this back into the equation of the ellipse for y
x^2 + (-2x)^2/4 = 1
x^2 + 4x^2/4 = 1
2x^2 = 1
x^2 = ±√(1/2)
When x = √(1/2), y = -2√(1/2)
When x = -√(1/2) , y = 2√(1/2)
Subbing into the equation of the line
-2√(1/2) = 2√(1/2) + b → b = - 4√(1/2)
And
2√(1/2) = -2√(1/2) + b → b = 4√(1/2)
See the graph here : https://www.desmos.com/calculator/7ifp0ajbri