First, we need to find the y-coordinates that correspond to x = 1. We plug x = 1 into x^2 y − 4x +
y^2 = 2, and rearrange a little, and we get y^2 + y − 6 = 0.Next, we factor the quadratic to get (y + 3)(y − 2) = 0, so we will be finding tangent lines at the
coordinates (1, −3) and (1, 2).
At (1, −3), we get = = = −2.
Therefore, the equation of the tangent line is y + 3 = −2(x − 1).
At (1, 2), we get = = .
Therefore, the equation of the tangent line is y = 2.