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Question

A tangent to the curve, y=f(x) at P(x,y) meets xaxis at A and yaxis at B. If AP:BP=1:3 and f(1)=1, then the curve also passes through the point.

A
(13,23)
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B
(3,128)
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C
(12,3)
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D
(2,18)
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Solution

The correct option is D (2,18)
(yy1)(xx1=f(x1)
yy1=f(x1)(xx1)
y=0,yf(x1)=xx1
x=x1=y1f(x1)
A=(x1y1f(x1),0)
x=0,yy1=f(x1)(x1)
y=y1x1f(x1)
B=(0,y1x1f(x1))
P divides AB in ratio 1:3

x1=3(x1y1f(x1))4
y1=y1x1f(x1)4
4y1=y1x1f(x1)
f(x1)=3y1x1
f(x)=3yx
dydx=3yx
dyy=3dxx
lny=3lnx+c
y=kx3
y(1)=1
k=1
y=1x3.
Option D satisfies the above function.

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