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Question

# In this exercise, use the equation of the tractrix $y=a \operatorname{sech}^{-1} \frac{x}{a}-\sqrt{a^2-x^2}, \quad a>0$.Let $L$ be the tangent line to the tractrix at the point $P$. If $L$ intersects the $y$-axis at the point $Q$, show that the distance between $P$ and $Q$ is $a$.

Solution

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Let the points as follows,

$\begin{array}{lcl} P&=& (&x_1,& y_1)\\ Q&=&(&0, &y_2) \end{array}$

Now the slope at point $P$ is

\begin{aligned} m&= \frac{y}{x}=\frac{\sqrt {(\overline{PQ})^2-x^2}}{x} \end{aligned}

Writing the equation of the line ,

\begin{aligned} y-y_1&=m(x-x_1)\\ y&=\frac{\sqrt {(\overline{PQ})^2-x^2}}{x}(x-x_1)+y_1 \end{aligned}

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