In general can you use the standard form y – β = m(x – α) of a straight line of gradient m , going through a point (α,β) in the case of finding equation of a tangent line drawn to a circle through a given point (α,β). What happens if one of the tangent lines is vertical and what is your suggestion to cover all situations?

Consider the circle given by the equation, x² +y² -2x – 4y – 4 = 0 , if you want to find equations of tangents drawn to the circle through the point ( 4, 0 ) usual method is to write it as y = m( x – 4) but using this you can find only one value for m that is 5/12 because one tangent is vertical. Now you can use the form b ( y- β ) = a (x – α ) , that gives equation of the tangent in this case as, by = a ( x – 4) and when you use the condition of radius of the circle is equal to perpendicular distance from centre of the circle to the tangent line gives, b ( 12a – 5b ) = 0 that is b =0 or a/b = 5/12 , here you can find both equations as x =4 and y = 5/12( x – 4).