A curve `y=f(x)` passes through the origin. Through any point `(x , y)` on the curve, lines are drawn parallel to the co-ordinate axes. If the curve divides the area formed by these lines and co-ordinates axes in the ratio `m : n ,` find the curve.

Explanation

Area `(OAPB)=xy rArr "Area"(OAPO) = underset(0)overset(x)intf(t) dt`

Area `(OPBO) = xy - underset(0)overset(x)intf(t) dt rArr ("Area" (OAPO))/("Area" (OPBO)) = m/n`
`n underset(0)overset(x)intf(t) dt = m(xy-underset(0)overset(x)intf(t)dt) rArr n underset(0)overset(x)intf(t)dt = mx f(x) - m underset(0)overset(x)intf(t)dt`
Differentiating w.r.t x
`nf(x)=mf(x)+mxfcancelc(x)-mf(x) rArr (f'(x))/(f(x)) = n/m = 1/x`
`f(x) = cx^(n//m)`
similarly `f(x) = cx^(m//n)`