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Given equation of curve = xy=6at

p=(2.,3)

O=(0,0)

slope m= 3/2

OP is  Perpendicular to AB

therefore m=-2/3

From,y=mx+c

where m=-2/3 & P=(2,3)

by substituting ,3=-4/3+c

c=13/3

by solving we get y=13/3 and x=13/2

Let A=(0,y) and B=(x,0)

AB is divided by P in the ratio λ : 1

therefore

(2,3)= λ (13/2) / λ +1  ,, 13/3 / λ +1

By equating,,

2=λ (13/2) / λ +1     3=13/3 / λ +1

on solving we get λ=4/9

The ratio in which P divides AB in the ratio 4:9

Hope it helps!!!