A curve $C$ has parametric equations $x=2+t+\frac{2}{t}$, $\qquad\qquad$ $y=2-t+\frac{2}{t}$

Question

A curve $C$ has parametric equations $x=2+t+\frac{2}{t}$, $\qquad\qquad$ $y=2-t+\frac{2}{t}$, where $t$Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $t$ and hence find the exact value of $t$ for which the tangent to the curve at $t$ is parallel to the $y$-axis. Find the value of $t$ for which the distance from the point $(1, 0)$ to the curve is the shortest possible.

Accepted Answer

(i) $\frac{\mathrm dy}{\mathrm dx}=-\frac{t^2+2}{t^2-2}$, with a vertical tangent at $t=-\sqrt2$ (ii) $t=-0.938$ (3 s.f.)

2013 ACJC P2 Q3 [Modified]

2013 ACJC P2 Q3 [Modified]

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