The common chord of the circle x2+y2+6x+8y−7=0  and a circle passing through the origin and touching the line y=x , always passes through the point:

Let other circle be x2+y2+2gx+2fy=0 . [through origin]This touches the line y=x .∴       |−g+f2|=g2+f2 ⇒g2+f2−2gf=2(g2+f2) ⇒g2+f2+2gf=0 ⇒(g+f)2=0  i.e., g=−f .∴   Other circle is x2+y2+2gx−2gy=0     …..(1)Common chord of circle (1) and x2+y2+6x+8y−7=0 (2g−6)x−(2g+8)y+7=0 i.e., g(2x−2y)−(6x+8y−7)=0 which always passes through the point of intersection of   2x−2y=0,6x+8y−7=0 i.e.,  (1/2,1/2) .