calculus of two variables

find the point on the curve

x^2 + xy + y^2 - 1 = 0

which is closest to the line x + y = 2

i differentiated implicitly to find the points where the gradient is equal to that of the line - (1,-1), (-1,1)
then worked out the perpendicular line from this tangent to the line x + y = 2

then got the points of intersection as (0,2) and (2,0)

is this correct?