Practice section 5 of 6

A curve has $x=3t-1/t$, $y=t+4/(3t)$. Find the exact value(s) of $t$ at which the gradient is $1/6$. [4]

The curve $C$ has $x=t^2(t+6)$, $y=t^2+t-6$, $t<0$.

(i) Find the equation(s) of the tangent to $C$ parallel to the y-axis. [3]

(ii) Sketch $C$, showing its axial intercepts. [2]

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

(i) Find its cartesian equation. [2]

(ii) Sketch $C$. [2]

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

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

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]

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

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

(i) Sketch the curve. [2]

(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]

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 $8$.