Trouble with this implicit diff/tangent line problem

Question

Find the equation of all the tangent lines to the curve x²y+4y=8 when y=1

I wouldn't have trouble if they had provided x and y, like many problems like this do, but I don't have an x to work with. Basically I got to dy/dx = -2xy/x²+4, and figured out there are two separate tangent lines with points (-2,1) and (2,1). How do I get the slope for each set of points so that I can write the equations of the two tangent lines?

Answer 1

Your points are correct.

Since x²y + 4y = 8, 2xy + x²(dy/dx) + 4(dy/dx) = 0

So, dy/dx(x²+4) = -2xy. Thus, dy/dx = -2xy / (x² + 4)

For the point (2,1), dy/dx = -4/8 = -1/2. So, the tangent line is y - 1 = -(1/2)(x - 2). y = -(1/2)x + 2

For the point (-2,1), dy/dx = 4/8 = 1/2. So, the tangent line is y - 1 = (1/2)(x + 2). y = (1/2)x + 2