Find Circle from Tangent Line and Two Points

Question

I have two points a,b and a line L.

I want an equation to find point c that is the center of the circle that touches a, b and L

Thanks.

Answer 1

Family of circles passing through two points is : $$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)+\lambda L'=0$$

Here $L'$ is equation of line passing through points $a,b$. Write equation of circle in this form. Apply condition that distance of centre from $L(\text{tangent})$ is equal to radius. Get $\lambda$

Answer 2

There is a similar question elsewhere giving this answer:

Construct CF being the perpendicular bisector of AB. Produce L to meet CF at F. Produce L (the other way) to meet CB produced at D Produce CA and mark point E so that CE = CD Join EF Using any point on CF construct a circle to touch EF, DF at X, Y Join FA, where it cuts the constructed circle nearest to A mark as A' Join XA' and YA' Construct lines AX' and AY' parallele to XA' and YA' respectively. Consrtruct perpendiculars from X' and Y' to CF which should meet as the centre of a circle passing through A, B and touching L at Y'