A tangent drawn to the curve y = f(x) at P(x, y) cuts the x and y axes at A and B, respectively, such that AP : PB = 1 : 3. If f(1) = 1 then the curve passes through `(k,(1)/(8))` where k is

Options

A. 1
B. 2
C. 3
D. 4

Answer

B

Explanation

`(3X)/(x)+(Y)/(y)=4` is equation of tangent at P(x, y) having slope `-(3y)/(x)`
`therefore" "(dy)/(dx)=-(3y)/(x) rArr x(dy)/(dx)+3y = 0`
`therefore" "(dy)/(y)=-3(dx)/(x)`
`rArr" "log y =- 3 log x + log c`
`rArr" "y=(c)/(x^(3))` which passes through (1, 1)
`rArr" "c = 1`
Equation of the curve is `y = (1)/(x^(3))`