## Related questions with answers

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

VerifiedAnswered 2 years ago

Answered 2 years ago

Step 1

1 of 2Let 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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