Can a curve be defined by the function of the angle between the tangent line and an axis?

Question

The answer to this question is something I should know...but without regular practice working mathematics the answer is falling into some shadow of my understanding.

I have a curve which lies entirely within a Cartesian plane. The horizontal and vertical axes are represented by X and Y, respectively. The only thing I know about the curve is that the angle $(\lambda)$ this curve makes with the X-axis at any given value of Y is defined by this function: $$\lambda=f(y)=\tan^{-1}(\frac{a}{y})$$

I need to determine a function which will yield values of X with respect to Y according to this curve.

Generic Curve Plot

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What I tried to do, and what I am unsure of, was to assume that the function which describes $\lambda$ is equivalent to derivative of the curve. Which would mean the equation of the curve could be found by: $$x=\int f(y)\ dy=\int \tan^{-1}(\frac{a}{y})=y\tan^{-1}(\frac{a}{y})+\frac{1}{2}a\ln(a^2+y^2)$$

Is this correct? Can the curve be entirely defined by the function of the tangent angle with respect to y?

I appreciate your help!

Answer 1

The function that describes $\lambda$ is closely related to the derivative of the curve - the derivative is simply the gradient of the tangent so $\lambda$ is simply $\tan^{-1} (\frac{dy}{dx})$. So your assumption that the angle is the derivative is not quite correct.

Answer 2

No, you shouldn't be integrating the angle with respect to $y$.

We have $$\tan\lambda=\frac{\mathrm{d}y}{\mathrm{d}x}$$ so $$ \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{a}{y}. $$