The combined equation of three lines is $\left(x^2-y^2\right)\left(2x+5y+7\right)=0$ ,the number of circles which touches all three lines is 

If the lines form a triangle, then the number of circles which touches all three lines is 4

If the lines do not form a triangle, and two of them are parallel, third one is not parallel then the number circles which touches three lines is 2

If the lines do not form a triangle, and three are parallel lines, then the number of circles which touches all the lines is zero. 

If the lines do not forms a triangle, and three are concurrent, then the number of circles which touches three lines is one.

Given lines are 

$\begin{array}{l}2x+5y+7=0----\left(1\right)\\ x-y=0----\left(2\right)\\ x+y=0-----\left(3\right)\end{array}$

Observing the above three lines, no two lines are parallel, and the point of intersection of (2) and (3) lines is origin, which does not lies on (1)

It implies that the given three lines forms a triangle, hence the number of circles which touches all three lines is 4