Calculus Level 5

A curve passes through \((1,0)\) such that the ratio of the square of the intercept cut by any tangent off the \(y\)-axis to the subnormal is equal to the ratio of the product of the co-ordinates of the point of tangency to the product of square of the slope of the tangent and the sub-tangent at the same point.If the equation of the curve is \(y=f(x)\) then \(y=(e/b)\) at \(x=e\). Then \(b\) equals?

Details and Assumptions:

  • There are two such possible curves of the form \(x=e^{\pm F(x,y)} \) where \(F(x,y)\) is a function involving \(x\) and \(y\). Use \( +F(x, y) \).
  • \(e\) is the mathematical constant, \( e=2.718\ldots \).

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