The tangent at a point P(x,y) on a curve meets the axes at M and N such that P divides MN internally in the ratio 2:1. Then find the equation of the curve?

let eqn of line L: x/a + y/b = 1

where a, b are X,Y intercepts of line L

so, dy/dx = -1/a

and the point P which divides MN internally in the ratio 2:1

so x=a/3 and y=2b/3

on substiting in diff. eqn , we have

dy/dx = -y/2x

i.e (1/y) dy  = -(1/2x) .dx

ln y = -1/2lnx +lnC where lnC is constant

so eqn. of curve is :  y=(kx)^-1/2