Math topic中文

Distance between two points

Distance between two points.

Practice questions

  1. 1The lines l1l_1 and l2l_2 have equations r=(4110)+λ(1−1−6),λ∈R\mathbf{r} = \begin{pmatrix} 4 \\ 1 \\ 10 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ -6 \end{pmatrix}, \lambda \in \mathbb{R}…
  2. 2A drilling company plans to install a straight pipeline ABAB through a mountain. Points (x,y,z)(x, y, z) are defined relative…
  3. 3在平面直角坐标系xOy中，已知A（2，-2），在y轴上确定点P，使△AOP为等腰三角形，则符合条件的点有 ( ) A. 2个 B. 3个 C. 4个 D. 5个
  4. 4已知线段MN/ ⁣/xMN/\!/x轴，点MM的坐标是(−3,5)(-3,5)，线段MNMN的长为66，则点NN的坐标是___.
  5. 5点M(0,2)M(0,2)与圆CC：(x−a)2+y2=1(a∈R)(x-a)^{2}+y^{2}=1(a\in R)的位置关系是()(\quad) A. 点MM在圆CC内 B. 点MM在圆CC上 C. 点MM在圆CC外…
  6. 6The equation of a curve is x2−2xy+2y2=20x^2 - 2xy + 2y^2 = 20. Find the equations of the tangent and normal to the curve at the poi…
  7. 7已知圆C1\mathbf{C}_{\mathbf{1}}：x2+y2-6x-6=0\mathbf{x}^{\mathbf{2}}\mathbf{\text{+}}\mathbf{y}^{\mathbf{2}}\mathbf{\text{-}6}\mathbf{x\text{-}6\text{=}0}…
  8. 8已知数轴上有AA，BB两点，点AA与点BB之间的距离为11，点AA与原点OO之间的距离为33，那么点BB对应的数是 .
  9. 9The plane pp passes through the points AA, BB and CC with coordinates (0,1,1)(0, 1, 1), (2,−1,4)(2, -1, 4) and (−2,−1,0)(-2, -1, 0) re…
  10. 10A curve has parametric equations x=2tx=2t, y=e−t2y=e^{-t^2}. Sketch the curve, stating the exact coordinates of any points of…
  11. 11The lines l1l_1 and l2l_2 have equations x−13=y−2a,z=1andr=(120)+λ(−111),λ∈R\frac{x-1}{3} = \frac{y-2}{a}, z = 1 \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}, \lambda \in \mathbb{R}…
  12. 12已知点(a+1,a−1)(a+1,a-1)在圆x2+y2−2ay−4=0x^{2}+y^{2}-2ay-4=0外，则实数aa的取值范围为()(\quad) A. a<1a< 1 B. a>1a>1 C. 0<a<10< a< 1 D. a>15a>\dfrac{1}{5}…
  13. 13Methane ( CH_4 ) is a chemical compound with a tetrahedral structure. The 4 hydrogen (H) atoms form a regular tetrahedro…
  14. 14在一张纸上有一圆CC：(x+2)2+y2=r(r>0)(x+2)^{2}+y^{2}=r(r>0)与点M(m,0)(m≠−2)M(m,0)(m≠-2)，折叠纸片，使圆CC上某一点M′M'恰好与点MM重合，这样的每次折法都会留下一条直线折痕PQPQ，设折痕…
  15. 15FF为抛物线CC：y2=4xy^{2}=4x的焦点，点M(m,4)M(m,4)在CC上，直线MFMF交CC的准线于点NN，则∣FN∣=()|FN|=(\quad) A. 54\dfrac{5}{4} B. 103\dfrac{10}{3}…
  16. 16已知：数轴上AA、BB两点表示的有理数为aa、bb，且(a−1)2+∣b+2∣=0(a-1)^{2}+|b+2|=0． (1)A(1)A、BB各表示哪一个有理数？ (2)点CC在数轴上表示的数是cc，且与AA、BB…
  17. 17已知圆OO：x2+y2=4x^{2}+y^{2}=4和圆CC：(x−2)2+(y−3)2=1.(x-2)^{2}+(y-3)^{2}=1.现给出如下结论，其中正确的是()(\quad) A. 圆OO与圆CC有四条公切线 B. 过CC且在…
  18. 18The curve CC has equation 2x−y2=(x+y)22x - y^2 = (x + y)^2. Find the equations of the tangents to CC which are parallel to the xx…
  19. 19Baby Peter is pulling a toy wagon on a flat ground which is modelled by the xyxy-coordinate plane. Peter is initially at…
  20. 20设m∈Rm\in R，过定点AA的动直线x+my=0x+my=0和过定点BB的动直线mx−y−m+3=0mx-y-m+3=0交于点P(x,y)P(x,y)，则∣PA∣+∣PB∣|PA|+|PB|的取值范围是()(\quad) A. [5,2 5][\sqrt{5},2\:\sqrt{5}]…
  21. 21A frigate is stationed at position F(1,2,0)F(1, 2, 0). Two submarines S1S_1 and S2S_2 are under the sea surface. Submarine S1S_1…
  22. 22The equation of a curve is given by 4x3+3x2y=y3−24x^3+3x^2y=y^3-2. Find dydx\dfrac{dy}{dx} in terms of xx and yy, simplifying your…
  23. 23The 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…
  24. 24过抛物线x2=4yx^{2}=4y在第一象限内的一点PP作切线，切线与两坐标轴围成的三角形的面积为12 \dfrac {1}{2}，则点PP到抛物线焦点FF的距离为()() A. 11 B. 22 C.…
  25. 25A 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.…
  26. 26在直角坐标系xOyxOy中，直线ll的参数方程为{y=−5+12tx=−32t(t \begin{cases} \overset{x=- \dfrac { \sqrt {3}}{2}t}{y=-5+ \dfrac {1}{2}t}\end{cases}(t为…