Discussion questions

Find the coordinates of the intersections of the curve

with the x- and y-axes. Sketch the curve.

A curve $C$ has parametric equations $x=t^3+2/t^3$, $y=t^3-2/t^3$, $t\in\mathbb R$. Find a Cartesian equation of $C$. Sketch it, giving exact axis intersections and equations of asymptotes.

Find the Cartesian equation for each curve and give restrictions on $x$ where possible.

(i) $x=e^t$, $y=e^{3t}$.

(ii) $x=\ln t$, $y=\sqrt t$, for $t>0$.

(iii) $x=5+4\sec\theta$, $y=1+\tan\theta$.

A curve has $x=3\cos\theta$, $y=2+\sin\theta$, $0\le\theta\le2\pi$.

(a) Show that the tangent at $P$, where $\theta=\pi/6$, is $y+(\sqrt3/3)x-4=0$.

(b) Find the normal at $P$.

(c) The tangent and normal meet the y-axis at $T$ and $N$ respectively. Find the exact area of triangle $PNT$.

Curve $C$ has $x=1/\sqrt{1+t^2}$, $y=t^3+1$, where $t\le0$.

(i) Find $dy/dx$ in terms of $t$.

(ii) Find the coordinates of $P$ where $C$ cuts $y=1/x^2-2$.

Curve $C$ has $x=3t^2$, $y=6t$. The tangent at $P=(3p^2,6p)$ meets the y-axis at $D$. Find the Cartesian equation of the locus of the midpoint of $PD$ as $p$ varies.

A curve has $x=3t+2$, $y=t+3/t$, where $t\ne0$. Find the tangent at $P$ where $x=11$, and determine whether that tangent cuts the curve again.8 marks

Curve $C$ has $x=\theta-\sin\theta$, $y=1-\cos\theta$, $0\le\theta\le2\pi$.

(i) Show that $dy/dx=\cot(\theta/2)$ and find the gradient at $\theta=\pi$. Describe the tangents as $\theta\to0$ and $\theta\to2\pi$.

(ii) Sketch $C$, clearly showing its features at $\theta=0,\pi,2\pi$.

(iii) For $P$ with parameter $p$, $0<p<\pi/2$, show that the normal at $P$ crosses the x-axis at $(p,0)$.