Find the points where the following graph has a vertical tangent line

Question

Find the points where the following graph has a vertical tangent line.

4x² + y² -8x + 4y + 4=0

Answer 1

Differentiate implicitly with respect to x,

8x + 2yy' - 8 + 4y' = 0

Solve for y',

y' = 4(1-x)/(y+2)

As y-->-2, y'-->+/-oo. So, the graph has a vertical tangent line.

Plug in y = -2 in the original equation to solve for x,

4x² + 4 - 8x - 8 + 4 = 0

4x(x-2) = 0

x = 0, 2 <==Answer