Tangents at a point

A tangent is perpendicular to the radius at its contact point. This is a geometric method; differentiation is unnecessary. Method: check the point satisfies the circle equation; join it to the centre; use the negative reciprocal gradient when both gradients are finite. Then use the point–gradient equation. Example: C=(0,0), P=(3,4) on x2+y2=25. Radius gradient 43, tangent gradient - 34, tangent 3x+4y=25. A horizontal radius gives a vertical tangent x=x_P; a vertical radius gives a horizontal tangent y=y_P. Exam check: do not divide by zero or use the radius gradient as the tangent gradient.