Finding tangents with unknown contact points

A tangent meets a circle once. For a line with an unknown constant or gradient, the substituted quadratic therefore has discriminant zero. Equivalently, the perpendicular distance from the centre to the line equals the radius. For Ax+By+C=0, the centre (a,b) has perpendicular distance d= |Aa+Bb+C|sqrt(A2+B2). To see why, the perpendicular direction is (A,B): a displacement t(A,B) changes the line expression by t(A2+B2) and has length |t|sqrt(A2+B2). Method: write a line through the supplied point, or a line with the required gradient. Apply the repeated-root or distance condition. Solve all parameter values and check separately whether a vertical tangent was omitted by y=mx+c. Example: tangents to x2+y2=9 parallel to y=x have form y=x+c. Then |c| 2=3, so c=+-3 2. Exam check: an exterior point normally has two tangents. A point on the circle has only one and an interior point has none. A vertical tangent has no finite gradient.