Find the equation of a circle which passes throught point (1,2) and touches the line y=x+3.The point (3,4) is inside the circle.

As you can see many circles are possible and the equation of these circles can be found out by taking a chord parallel to y=x+3 starting with 1,2 and beyond 3,4 and finding the circum circle of end point of these chords and a symmetric point on the line y=x+3

Hi friend

You can see that more than one circles cannot possess this condition.If you try pass pass one more circle having the same point inside itself it will intersect the line at two points.You can try to draw figure on the paper as I have drawn.If you see more than two circles then you can repost your reply.