The common  chord of the circle $x^2+y^2+6x+8y-7=0$  and a circle  passing through  the  origin  and  touching     the  line y=x  always  passes through the point  

Let  the second circle  be  $x^2+y^2+2gx+2fy=0$
But  y=x  touches the circle .
Hence , $x^2+x^2+2gx+2fy=0$ has equal  roots ,f+g=0 There  fore ,the  equation  of  the common chord  is  $2(g-3)x+2(-g-4)y+7=0$
Which  passes through  the point  of  intersection  of  $-6x-8y+7=0and2x-2y=0$