This is a good observation which allows an alternative solution (not the way where you pick two generic points, minimizing the distance).

The given line has slope -1, so the normal line has slope 1, making it of the form y = x + k. Now you want to know the point on the ellipse where this line intersects it perpendicularly.

Implicit differentiation yields that the tangent direct is -x/(4y). This is normal to the slope 1 if their product is -1, so: -x/(4y) = -1 <=> x = 4y. Substitution in the equation of the ellipse now gives a quadratic equation in x or y.