Need help with alternative method to equation of a tangent at the point of a circle

Question

so I know a simpler of looking for the equation of a tangent at the point of a circle is to differentiate, my lecturer would rather we not use calculus and has charged us with looking for an alternate method. Any pointers guys?

Answer 1

Find the slope of the line passing through the center of the circle and the point on the circle. Call it $m$. Then form the equation of a line through the point on the circle but let the slope of this line be the opposite reciprocal of the one you just found, i.e. slope $\frac{-1}{m}$.

Answer 2

Hint: The tangent line at a point on a circle is perpendicular to the diameter of the circle passing through that point

Also, if a line has slope $s$, another line perpendicular to the first has slope $-1/s$