Finding the center of a circle Let us suppose there are two straight lines having equation x=5 and y=7 and a circle is drawn such that these two straight lines are tangents to the circle. Now we are required to find the center of the circle. How to find it? I mean aren't there four different circles possible? Which circle's center is the question asking about?

It is not possible to find the center of the circle, because (as mentioned in the comments before) there is an infinite amount of possible circles.  But all of them have some property in common:


*

*The center lies either on the line defined by $\left\lbrace (x,y) : (5,7)+r\cdot(1,1), r\in\mathbb{R}\right\rbrace$
or

*The center lies on the line defined by $\left\lbrace (x,y) : (5,7)+r\cdot(1,-1), r\in\mathbb{R}\right\rbrace$


If $p$ is point on one of those lines, then radius of this circle with center $p$ is given by
$$\frac{1}{\sqrt{2}}\left\|p-(5,7)\right\|$$
