The circles x2+y2+2x+4y−20=0 and x2+y2+6x−8y+ 10 = 0

r1=5;r2=15;C1C2=40 ∴ r1+r2>C1C2>r1−r2 Hence, the circles intersect at two distinct points. There are two common tangents. Also, 2g1g2+2f1f2=2(1)(3)+2(2)(−4)=−10  and c1+c2=−20+10=−10 Thus, the two circles are orthogonal  Length of common chord =2r1r2r12+r22=532  Length of common tangent =C1C22−r1−r22=51251/4.