Calculus-implicit differentiation

The normals to the curve that are parallel to the line 2x + y = 0 are those normals which are perpendicular to the gradient of the line, which is the vector (2,1). Therefore, we can use implicit differentiation to find the gradient of our curve: y' = (2x + 2 - 1)/(-x) The gradient of the normal is -1/2, so the normal equation is: y = -(x/2) + k where k is a constant. Therefore, the two normals to the curve that are parallel to the line 2x + y = 0 are given by: y = - (x/2) + k1 and y = - (x/2) + k2 where k1 and k2 are two constants.