Answer
Any tangent line at a point $P$ is perpendicular to the radius $OP$
Work Step by Step
Let the center of the circle be $O=(a,b)$ and let the radius be $r$
Then the equation for the circle is: $(x-a)^2+(y-b)^2 = r^2$
We can find $y'$:
$2(x-a)+2(y-b)~y' = 0$
$y' = -\frac{x-a}{y-b}$
A line that is tangent at the point $P=(x_0,y_0)$ has the slope $m_t = -\frac{x_0-a}{y_0-b}$
We can find the slope $m_r$ of the radius $OP$:
$m_r = \frac{y_0-b}{x_0-a}$
Since $m_t = -\frac{1}{m_r}$, the tangent line is perpendicular to the radius $OP$
