Review practice · Section 2 of 2

The curve $C$ has $x=2\cos t$, $y=\sin t$, $0\le t<2\pi$.

(i) Find its cartesian equation.2 marks

(ii) Sketch $C$.2 marks

(iii) Show that the tangent at the point with parameter $t$ is $(\cos t)x+(2\sin t)y=2$.3 marks

(iv) Hence find the tangents at $P$, where $t=7\pi/6$, and $Q$, where $t=11\pi/6$.2 marks

The curve $C$ has $x=k\cos t$, $y=k\sin2t$, $-\pi/4\le t<\pi/4$, where $k>0$.

(i) Find the normal at $(k\cos p,k\sin2p)$, where $-\pi/4<p<\pi/4$.3 marks

(ii) Deduce the tangent at $t=0$.2 marks

A curve has $x=\cos^2t$, $y=\sin^3t$, $0\le t\le\pi/2$.

(i) Sketch the curve.2 marks

(ii) The tangent at $(\cos^2\theta,\sin^3\theta)$ meets the x- and y-axes at $Q$ and $R$ respectively. With $O$ the origin, show that the area of triangle $OQR$ is $\tfrac1{12}\sin\theta(3\cos^2\theta+2\sin^2\theta)^2$. The source repeats the interval using $t$; the tangent parameter is $\theta$.6 marks

A curve is given by $x=at^2$, $y=2at$.

Show that the normal at $(at^2,2at)$ is $y+tx=at^3+2at$.

If that normal meets the curve again at $(aq^2,2aq)$, show that $t^2+qt+2=0$. Deduce that $q^2$ cannot be less than $8$.

The curve $C$ has $x=3/t$, $y=2t$.

(i) The line $y=2x$ cuts $C$ at $A$ and $B$. Find the exact length of $AB$.3 marks

(ii) The tangent at $P=(3/p,2p)$ meets the x-axis at $D$ and the y-axis at $E$. Let $F$ be the midpoint of $DE$. Find the cartesian equation of the curve traced by $F$ as $p$ varies.5 marks