Show that the tractrix is orthogonal to certain half-circles

Show that the tractrix discussed in Example 1.17 is orthogonal to the lower half of each circle with radius $a$ and center on the positive $y$-axis.

Example 1.17:

$$\left(y^{\prime}\right)^2=x^2y^{\prime\prime}$$

We note that $y$ is missing, so we make the substitutions $p=y^{\prime}$ and $p^{\prime}=y^{\prime\prime}$.

Thus the equation becomes:

$$p^2=x^2p^{\prime}$$

Now use separation of variables:

$$\frac{dx}{x^2}=\frac{dp}{p^2}$$

This integrates to:

$$-\frac{1}{x}=-\frac{1}{p}+E$$

for some unknown $E$. We resubstitute $p=y^{\prime}$ and integrate to obtain:

$$y(x)=\frac{x}{E}-\frac{1}{E^2} \ln(1+Ex)+D$$

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 I don't see a question mark anywhere. – joriki Sep 21 '12 at 13:39