Find a tangent point on circle?

Question

Given a line with first end point P(x1,y1) another end point is unknown, intersect with a circle that located at origin with radius R at only one point(tangent) T(x2,y2). An

Answer 1

Take R as the radius of the circle and D the distance from the external point to the center of the circle such that D > R.

The tanget line makes and angle of \alpha with the line connecting the external point and the center, where

\alpha = arcsin(R/D)

The line connecting the external point (P) and the center (C) makes an angle with the horizontal of

\beta = arctan((C_y - P_y)/(C_x - P_x))

That gives you the angle of the tangent line with the horizontal as

\theta = \beta +/- \alpha

Note the ambiguity.

The length of the tangent segment is

L = sqrt(D^2 - R^2)

which is all you need.