Math topic中文

Parametric equations

Simple parametric equations and their graphs.

Practice questions

  1. 1A curve CC has parametric equations x=2cos⁡θ+cos⁡2θ,y=2sin⁡θ−sin⁡2θ,x = 2\cos\theta + \cos 2\theta, \quad y = 2\sin\theta - \sin 2\theta, where 0<θ<2π0 < \theta < 2\pi…
  2. 2The parametric equations of a curve C are x=at,  y=at3x = at,\; y = at^{3}, where aa is a positive constant. The point P on the c…
  3. 3选修4−44-4：坐标系与参数方程 在直角坐标系xOyxOy中，曲线C1C_{1}的参数方程为{x=3cos⁡α,y=sin⁡α(α\begin{cases}x= \sqrt{3}\cos α, \\ y=\sin α\end{cases}(α为参数).).…
  4. 4The curve CC is defined by the parametric equations x=t−sin⁡tx = t - \sin t, y=sin⁡2ty = \sin^{2} t,   0≤t≤π\;0 \le t \le \pi. The point…
  5. 5A curve C is defined by the parametric equations x=θ2−1,y=tan⁡θ2,0≤θ<πx = \theta^2 - 1,\quad y = \tan\frac{\theta}{2},\quad 0 \le \theta < \pi…
  6. 6在平面直角坐标系xOyxOy中，曲线CC的参数方程为{y=4sin⁡ax=4cos⁡a+2(a \begin{cases} \overset{x=4\cos a+2}{y=4\sin a}\end{cases}(a为参数))，以OO为极点，以xx轴…
  7. 7A curve CC is given parametrically by the equations x=2cos⁡3θx = 2\cos^3 \theta, y=2sin⁡3θy = 2\sin^3 \theta where −π2<θ<π2-\frac{\pi}{2} < \theta < \frac{\pi}{2}…
  8. 8若直线{x=1−ty=2+3t(t\begin{cases}{x=1-t}\\ {y=2+3t}\end{cases}(t为参数))与直线y=kx−1y=kx-1平行，则常数k=()k=(\quad) A. −3-3 B. −13-\dfrac{1}{3}…
  9. 9The points (x,y)(x,y) on a curve satisfy the equations x=12t2−2tx = \tfrac{1}{2}t^2 - 2t and dydx=12t−1\dfrac{dy}{dx} = \dfrac{1}{2t - 1}…
  10. 10A curve CC has parametric equations x=3t2,y=6t3,where t is a real parameter.x = 3t^2, \quad y = 6t^3, \quad \text{where } t \text{ is a real parameter.} Sk…
  11. 11The curve C1C_1 has equation y=ax2+bx+3x+2y = \frac{ax^2+bx+3}{x+2}, where aa and bb are constants. It is given that C1C_1 has an…
  12. 12A curve C is defined by the parametric equations x=cos⁡θ−1cos⁡θ,  y=cos⁡θ+1cos⁡θ,  −π2<θ<π2.x = \cos\theta - \dfrac{1}{\cos\theta},\ \ y = \cos\theta + \dfrac{1}{\cos\theta},\ \ -\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}.…
  13. 13A curve C1C_1 is defined parametrically by x=2t−1andy=4t+1,t≠±1.x = \frac{2}{t-1} \quad\text{and}\quad y = \frac{4}{t+1},\qquad t \ne \pm 1.…
  14. 14A curve CC is defined parametrically by x=atan⁡t,y=asec⁡2tsin⁡t,−π4<t<π4,x = a \tan t, \quad y = a \sec^2 t \sin t, \quad -\frac{\pi}{4} < t < \frac{\pi}{4},…
  15. 15A farmer has a plot of the land, OABOAB, with adjacent sides OAOA and OBOB. With reference to OO as the origin, OAOA and…
  16. 16[Cambridge/2003] Sketch the curve with parametric equations x=3t,y=3t.x = 3t, \quad y = \frac{3}{t}. The point PP on the…
  17. 17在平面直角坐标系中，曲线C1C_{1}的参数方程为{x=−1+5cos⁡αy=−2+5sin⁡α(α\begin{cases}x=-1+\sqrt{5}\cos α\\ y=-2+\sqrt{5}\sin α\end{cases}(α为参数).).以坐标原点OO为极点…
  18. 18The parametric equations of a curve C are x=2acos⁡(t+π6),y=asin⁡t,x=2a\cos\left(t+\tfrac{\pi}{6}\right),\quad y=a\sin t, where 0≤t<2π0\le t<2\pi…
  19. 19The curve CC has parametric equations x=2t−1t2,y=2t+1t,t∈R,t≠0.x = 2t - \frac{1}{t^2}, \quad y = 2t + \frac{1}{t}, \quad t \in \mathbb{R}, t \ne 0.…
  20. 20The curve CC is defined by the parametric equations x=t2x = t^2 and y=2t+t3y = 2t + t^3, where −1≤t≤1-1 \le t \le 1. Sketch the gr…
  21. 21A curve has parametric equations x=2tx=2t, y=e−t2y=e^{-t^2}. Sketch the curve, stating the exact coordinates of any points of…
  22. 22Projectile motion is the motion of a launched object, subject to only the constant acceleration of gravity, g ms−2g\text{ ms}^{-2}…
  23. 23The curve C1C_1 has equation y=ax2+bx+3x+2y = \dfrac{ax^2+bx+3}{x+2}, where aa and bb are constants. It is given that C1C_1 has a…
  24. 24把下列参数方程化为普通方程，并说明它们各表示什么曲线： (1){x=5cos⁡φ,y=4sin⁡φ(φ(1)\begin{cases} x=5\cos φ, \\ y=4\sin φ \end{cases}(φ为参数))； (2){x=1−3t,y=4t(t(2)\begin{cases} x=1-3t, \\ y=4t \end{cases}(t…
  25. 25A complex number zz varies with tt such that z=2cos⁡t+i(3sin⁡t),where 0≤t<2π.z = 2\cos t + \mathrm{i}(3\sin t), \quad \text{where } 0 \le t < 2\pi.…
  26. 26An astroid is a curve with parametric equations x=acos⁡3t, y=asin⁡3t.x = a \cos^3 t, \ y = a \sin^3 t. Sketch an astroid with a=2a = 2 a…
  27. 27The curve C is defined by the parametric equations x=a(1+1t)x = a\left(1 + \dfrac{1}{t}\right) and y=a(t−1t2)y = a\left(t - \dfrac{1}{t^2}\right)…
  28. 28The curve CC has parametric equations x=sin⁡θ+1andy=3cos⁡θ−1, where −π2≤θ≤π2.x = \sin \theta + 1 \quad \text{and} \quad y = \sqrt{3} \cos \theta - 1, \text{ where } -\frac{\pi}{2} \le \theta \le \frac{\pi}{2}.…
  29. 29The diagram below shows a region RR bounded by the curve (y+5)2=x−3(y+5)^{2}=x-3 and the line y=x−10y=x-10. Find the volume of soli…
  30. 30A curve CC is defined by the parametric equations x=2cos⁡θ+1,y=sin⁡θ,where −π≤θ≤π.x = 2\cos\theta + 1, \quad y = \sin\theta, \quad \text{where } -\pi \le \theta \le \pi.…
  31. 31The parametric equations of a curve CC are x=t+2et,y=2t+e−t,t>−2.x = t + 2e^t, \quad y = 2t + e^{-t}, \quad t > -2. Sketch CC. Find the…
  32. 32The graph of y=f(x)y = f(x) is given below. On separate diagrams, sketch the graphs of y=1f(x)y = \frac{1}{f(x)} y=f′(x).y = f'(x). A c…
  33. 33在平面直角坐标系xOyxOy中，以原点OO为极点，xx轴的正半轴为极轴建立极坐标系，曲线C1{{C}_{1}}的方程为x22+y2=1\dfrac{{{x}^{2}}}{2}+{{y}^{2}}=1，曲线C2{{C}_{2}}…
  34. 34在直角坐标系xoyxoy中，直线ll过点M(3,4)M(3,4)，其倾斜角为45∘45^{\circ}，以原点为极点，以xx正半轴为极轴建立极坐标，并使得它与直角坐标系xoyxoy有相同的长度单位，圆CC的极坐标方程为ρ=4sin⁡θρ=4\sin θ…
  35. 35A curve CC is defined parametrically by x=atan⁡t,y=asec⁡2tsin⁡t,−π4<t<π4,x = a\tan t, \qquad y = a\sec^{2}t\sin t, \qquad -\frac{\pi}{4} < t < \frac{\pi}{4},…
  36. 36In an experiment to test the design theory of launching a new space telescope into space, scientists monitor the movemen…
  37. 37The parametric equations of a curve are x=t+2x = t + 2, y=t2t−1y = \dfrac{t}{2t - 1}, where t∈Rt \in \mathbb{R}, t≠12t \ne \dfrac{1}{2}…
  38. 38A curve CC has parametric equations x=t−at,y=t+at, where t∈R, t≠0 and a is a positive constant.x = t - \frac{a}{t}, \quad y = t + \frac{a}{t}, \text{ where } t \in \mathbb{R}, \ t \ne 0 \text{ and } a \text{ is a positive constant.}…
  39. 39A curve CC is defined by the parametric equations x=t2,y=t3.x = t^2, \quad y = t^3. Prove that the equation of the tangent at…
  40. 40A curve CC has parametric equations x=e2t, y=t2.x = \mathrm{e}^{2t}, \ y = t^2. Find the value of tt at the point where the ta…
  41. 41A curve CC is defined by the parametric equations x=4sin⁡t,y=bcos⁡t−1,\begin{aligned} x &= 4\sin t, \\ y &= b\cos t - 1, \end{aligned}…
  42. 42A curve CC has parametric equations x=1−2cos⁡2θ,y=sin⁡2θ−1,for 0≤θ≤3π8.x = 1 - 2\cos 2\theta, \quad y = \sin 2\theta - 1, \quad \text{for } 0 \le \theta \le \frac{3\pi}{8}.…
  43. 43A curve C1C_1 has parametric equation x=3sec⁡2θx = 3\sec^2 \theta, y=2tan⁡θy = 2\tan \theta, 0≤θ<π20 \le \theta < \frac{\pi}{2}. L1L_1 an…
  44. 44Max Min & rate Of Change Problem not tested 2022 H2 MA CT. In an experiment to test the design theory of launching a…
  45. 45A complex number zz varies with tt such that z=2cos⁡t+i(3sin⁡t),where 0≤t<2π.z = 2\cos t + i(3\sin t),\quad \text{where } 0 \le t < 2\pi. By takin…
  46. 46The diagram shows the curve y=f(x)y = f(x), with turning points at A(3,2.5)A(3, 2.5) and B(0,1)B(0, 1). The lines y=1y = 1 and x=−2x = -2…
  47. 47A curve CC has parametric equation x=t2−1,y=4t−t3+15,x = t^2 - 1,\qquad y = 4t - t^3 + 15, where t∈Rt \in \mathbb{R}. Show that the cu…
  48. 48A curve C1C_1 is defined parametrically by x=2t−1andy=4t+1,t≠±1.x = \frac{2}{t-1}\quad\text{and}\quad y = \frac{4}{t+1},\qquad t \ne \pm 1.…
  49. 49A curve C has parametric equations x=et,  y=sin⁡tx = e^{t},\ \ y = \sin t where 0≤t≤π0 \le t \le \pi. Find ∫etsin⁡t dt\int e^{t}\sin t\,dt . Ske…
  50. 50在平面直角坐标系xOyxOy中，以OO为极点，xx轴非负半轴为极轴建立坐标系，曲线MM的极坐标方程为ρ=4cos⁡θρ=4\cos θ，直线ll的参数方程为{y=tsin⁡αx=m+tcos⁡α(t \begin{cases} \overset{x=m+t\cos \alpha }{y=t\sin \alpha }\end{cases}(t…
  51. 51The curve C has parametric equations x=2t−1t2,y=2t+1t,t∈R, t≠0.ThepointPonthecurvehasparameterx = 2t - \dfrac{1}{t^{2}},\quad y = 2t + \dfrac{1}{t},\quad t \in \mathbb{R},\ t \ne 0. The point P on the curve has parameter …
  52. 52The parametric equations of a curve C1C_1 are x=3t3+2t2x=3t^3+2t^2 and y=−t3+2t2+1y=-t^3+2t^2+1 for −1≤t≤1-1 \le t \le 1. Another curve C2C_2…
  53. 53[Cambridge/9709/2010] The parametric equations of a curve are x=1+ln⁡(t−2),y=t+9t,for t>2.x = 1 + \ln(t - 2), \quad y = t + \frac{9}{t}, \quad \text{for } t > 2.…
  54. 54The curve CC has equation y=ln⁡(2−∣x∣)y=\ln\left(2-\lvert x\rvert\right). The region RR is bounded by CC, the lines y=ln⁡2y=\ln 2,…
  55. 55The curve CC has parametric equations x=5asec⁡θ,y=3atan⁡θ,where −π2<θ<π2x = 5a\sec\theta, \quad y = 3a\tan\theta, \quad \text{where } -\frac{\pi}{2} < \theta < \frac{\pi}{2}…
  56. 56A curve CC has parametric equations x=2sin⁡2t,y=cos⁡2t,for 0≤t<π.x = 2 \sin 2t, \quad y = \cos 2t, \quad \text{for } 0 \le t < \pi. Show that th…
  57. 57A curve CC has parametric equations x=(a2−t2)12,y=ln⁡t,for 0<t≤a and a>1.x = (a^2 - t^2)^{\frac{1}{2}}, \quad y = \ln t, \quad \text{for } 0 < t \le a \text{ and } a > 1.…
  58. 58A curve CC has parametric equations x=3t3, y=31+t2,t∈R.x = 3t^3,\ y = \frac{3}{1+t^2},\quad t \in \mathbb{R}. Find dydx\frac{\text{d}y}{\text{d}x}…
  59. 59The curve CC is defined by the parametric equations x=1−cos⁡t,y=sin⁡2t,where 0≤t≤π2.x = 1 - \cos t, \quad y = \sin 2t, \quad \text{where } 0 \le t \le \frac{\pi}{2}.…
  60. 60A complex number z varies with t such that z=2cos⁡t+i(3sin⁡t),where 0≤t<2π.z = 2\cos t + i(3\sin t),\quad\text{where } 0 \le t < 2\pi. By taking x=Re(z)x = \mathrm{Re}(z)…