Let P(34,512) divide AB internally in the ratio m:n.
Using the section formula, we get
(34,512)=(2m−n2m+n,−5m+32nm+n)
[∵ By internal section formula, the coordinates of point P dividing the line segment joining the point (x1,y1) and (x2,y2) in the ratio m1:m2 internally is
(m2x1+m1x2m1+m+2,m2y1+m1y2m1+m+2)]On equating, we get
34=2m−n2m+n and 512=−5m+32nm+n⇒34=4m−n2(m+n) and 512=−10m+3n2(m+n)⇒32=4m−nm+n and 56=−10m+3nm+n⇒3m+3n=8m−2n and 5m+5n=−60m+18n⇒5n−5m=0 and 65m−13n=0⇒n=m and 13(5m−n)=0⇒m =n does not satisfy.Since, m =n does not satisfy.∴5m−n=0⇒5m=n∴mn=15Hence, the required ratio is 1:5.