1. A curve has parametric equations $$x=4at^2 \quad y=a(2t+1)$$ where $a$ is a constant. Given that the curve passes through $(4,0)$, find $a$
$a=4$
2. A curve has parametric equations $$x = t+1 \quad y = t^2-1$$ Find the coordinates of the points of intersection of the curve and the line $x+y=6$.
$(-2,8)$ and $(3,3)$
3. Find and simplify an expression for $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $t$ in each of the following:
$x = 3t-1 \quad y = 2 - \dfrac{1}{t}$
$x = \sin^2 t \quad y = \cos^3 t$
$\dfrac{1}{3t^2}$
$-\dfrac{3}{2}\cos t$
4. A curve has parametric equations $$x = 4t-1 \quad y=\dfrac{5}{2t} + 10$$ Find a Cartesian equation for the curve.
$y=\dfrac{10}{x+1}+10$
5. A curve has parametric equations $$x=\cos t \quad y = \cos 2t$$ for $0 \leq t \leq \pi$.
Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ in its simplest form.
Find a Cartesian equation for the curve.
$4\cos t$
$y = 2x^2-1$
6. Find an equation to the normal of the curve with parametric equations $$x = 1-\cos2\theta \quad y = \sin2\theta$$ at the point where $\theta = \dfrac{\pi}{6}$.
$y = \sqrt{3}(1-x)$
7. A curve has parametric equations $$x = 3t^2 \quad y = 6t$$ The point $P$ has $y$ coordinate $-2$. The tangent and normal to the curve at $P$ meet the $x$ axis at the points $Q$ and $R$. Find the area of triangle $PQR$.
$\dfrac{20}{3}$
8. A curve has parametric equations $$x = \cos2\theta \quad y = \sin\theta\cos\theta$$ Find a Cartesian equation for the curve, and sketch it.
$x^2+4y^2 = 1$. Ellipse, centre origin, passes through $(\pm1,0)$ and $(0,\pm 0.5)$
9. Find the Cartesian equation for these parametric equations, and give their domains and ranges. $$x = \sqrt{t}-1 \quad y = 3\sqrt{t} \quad t > 0$$
$y=3x+3$ $x > -1$ and $y > 0$
10. A curve has equation $x^2 + y^2 = 9x-4$. A second curve has parametric equations $$x=t^2 \quad y = 2t$$ Find the coordinates of the intersection of these curves.
$(1,\pm2)$ and $(4,\pm4)$
11. A curve has parametric equations $$x = t^2 \quad y = \dfrac{6}{t}$$
Find an equation for the tangent to the curve at the point $(4, -3)$.
Find the value of $t$ when the tangent meets the curve again.
$3x-8y=36$
$t=4$
12. A curve has parametric equations $$x = \sin\theta \quad y = \sin\left(\theta+\dfrac{\pi}{6}\right)$$ where $-\dfrac{\pi}{2} \leqslant \theta \leqslant \dfrac{\pi}{2}$. Find the Cartesian equation of the curve.
$y = \dfrac{1}{x-2}$ $x\in\mathbb{R}$, $x\neq2$ and $y\in\mathbb{R}$, $y\neq0$
$y = \dfrac{1}{9}(x-2)(x+7)$ $-13 < x < 11$ and $-\dfrac{9}{4} < y < 18$
14. Find the Cartesian equation for these parametric equations and sketch it: $$x = 2\sin t - 1 \quad y = 5\cos t + 4; \quad t\in\mathbb{R}$$
$25(x+1)^2 + 4(y-4)^2 = 100$ ellipse, centre $(-1,4)$, $4$ wide and $10$ tall
15. Show that the curve with parametric equations $$x=\dfrac{1}{3}\sin t \quad y = \sin 3t; \quad 0 < t < \dfrac{\pi}{2}$$ has Cartesian equation $$y=ax(1-bx^2)$$ and state its domain and range.
$y = 9x(1-12x^2)$ $0 < x < \dfrac{1}{3}$ and $-1 < y \leqslant 1$
16. Find an equation for the tangent of this curve at $t = 3$: $$x = \ln(4-t) \quad y = t^2 - 5$$
$y=-6x+4$
17. Find an equation for the normal of this curve at $t = \dfrac{\pi}{6}$: $$x = \sin 2t \quad y = \sin^2t$$
$y=-\dfrac{2\sqrt{3}}{3}x+\dfrac{5}{4}$
18. The curve $C$ has parametric equations $$x = 9\cos t - 2 \quad y = 9\sin t + 1$$ where $-\dfrac{\pi}{6} \leq t \leq \dfrac{\pi}{2}$
Find a Cartesian equation for the curve.
Sketch the curve.
Find the length of $C$.
$(x+2)^2+(y-1)^2 = 81$
See video
$6\pi$
19. Show that the line with equation $y=2x-5$ does not intersect the curve with parametric equations $$x=2t \quad y=4t(t-1)$$
$4t^2-8t+5=0$, discriminant is less than zero so no solutions
20. The position at time $t$ of a ball thrown in the air can be modelled by $$x=2t \quad y = -4.9t^2+10t$$ where $x$ is the horizontal distance travelled and $y$ is its height above ground.
Find the time the ball is in the air for.
Find the maximum height of the ball.
$\dfrac{100}{49}$
$\dfrac{250}{49}$
21. The position, in metres, at time $t$ minutes, of a theme park ride is modelled by $$x = 120\sin 10t \quad y = 120-120\cos 10t$$
Find a Cartesian equation for the motion of the ride.
Find the average speed of the ride in metres per minute.
$x^2 + (y-120)^2 = 120^2$
$$1200$
22. The position of a paper aeroplane in metres at time $t$ seconds is modelled by $$x = 18t \quad y = -4.9t^2 + 4t + 10$$ where $0 \leqslant t \leqslant k$ and where $x$ is the horizontal distance travelled and $y$ is its height above ground.
Find the initial height of the aeroplane.
Find $k$.
Find the maximum height of the aeroplane.
$10$
$1.89$
$10.8$
23. A curve has parametric equations $$x = a\sec\theta \quad y = b\tan\theta$$ where $a$ and $b$ are positive constants. Find an equation for the tangent to the curve at the point $\theta = \dfrac{\pi}{4}$ in terms of $a$ and $b$.
$ay = b(\sqrt{2}x-a)$
24. A curve $C$ has parametric equations $$x = \dfrac{3}{t} + 2 \quad y = 2t - 3 - t^2$$ where $t\in\mathbb{R}$, $t \neq 0$
Find a Cartesian equation for $C$ in the form $$y = \dfrac{ax^2+bx+c}{(x-2)^2}$$
