Let A be a point on the curve C: $\displaystyle {x}^{ 2} + {y}^{ 2} - 2x + 4 = 0$. If the tangent line at A passes through P(4,3), then the length of AP is?

So i'm stuck midway at the solution for this one.

So far I've gotten these equations

$\displaystyle {y}^{ '} = \frac{1-x}{ y} = m $ ; the tangent line slope at any point on the curve (by implicit differentiation)

$\displaystyle m = \frac{y-3}{x-4} $; the slope at point P; acquired by y-k=m(x-h)

and when I put those two equations together I get:

$\displaystyle {-x}^{ 2} + 5x - 4 + 3y= {y}^{ 2} $

And when I input this equation back into the original curve i end up with

x= -y

I'm stuck at this point because when I plug in the last equation back to original curve again, I'm getting an complex solution.

Help please!

Oh Gosh. I am sorry. That's supposed to be

$\displaystyle {x}^{ 2} + {y}^{ 2} - 2x - 4 = 0$