Which of the following is true about the curve x^2 - xy + y^2 = 3 at the point (2,1)?

Question

all of these are different answers, only one can be right.

a: dy/dx exists at (2,1) but there is no tangent line at that point

b; dy/dx exists at (2,1) , and the tangent line at that point is horizontal

c; dy/dx exists at (2,1), and the tangent line at that point is neither horizontal nor vertical

d: dy/dx does no exists at (2,1) and the tangent line at that point is vertical.

Answer 1

x² - xy + y² = 3

2x - y - x(dy/dx) + 2y(dy/dx) = 0

(dy/dx)(-x + 2y) = y - 2x

dy/dx = (y - 2x) / (-x + 2y)

At the point (2,1), dy/dx = -3 / 0, so dy/dx is undefined.

Therefore, the tangent line to the curve is vertical at the point (2,1)