Equation of a circle with centre (4, 3) touching the circlex2+y2=1, is

Let the circle be x2+y2−8x−6y+k=0 which touches the circle x2+y2=1. The equation of the common tangent to these circles is given by  S1−S2=0⇒8x+6y−1−k=0This is a tangent to the circle x2+y2=1∴ ±1=k+182+62⇒k2+1=±10⇒k=−11 or, 9Hence, the equations of the circles arex2+y2−8x−6y+9=0and,  x2+y2−8x−6y−11=0