Math topic中文

Circles

coordinate geometry of circles in the form:

  • (x−a)2+(y−b)2=r2(x - a)^{2} + (y - b)^{2} = r^{2}
  • x2+y2+2gx+2fy+c=0x^{2} + y^{2} + 2gx + 2fy + c = 0
    (excluding problems involving two circles)

Practice questions

  1. 1The curve CC has equation y=x2+x+3x−2y = \frac{x^2 + x + 3}{x - 2}. Without using a calculator, find the set of values that yy…
  2. 2Find the gradient function of the curve y=2x2−ln⁡x+cy = 2x^2 - \ln x + c. The curve touches the positive xx-axis at point TT. Fi…
  3. 3The radius of a circle increases at a rate of 3 cm s^{-1}. Find, in terms of π\pi, the rate of increase of the area whe…
  4. 4定积分∫01(2x−x2)dx\int_{0}^{1}(\sqrt{2x{-}x^{2}})dx等于()({ }) A. π4\dfrac{\pi}{4} B. π2\dfrac{\pi}{2} C. π−14\dfrac{\pi{-}1}{4}…
  5. 5已知两定点M(0,1)M(0,1)，N(1,2)N(1,2)，平面内一动点PP到MM的距离与PP到NN的距离之比为2 \sqrt {2}，直线y=kx−1y=kx-1与点PP的轨迹交于AA，BB两点． (…
  6. 6The equation of a circle is x2+y2+8x−20y−53=0x^2 + y^2 + 8x - 20y - 53 = 0. Find the radius of the circle and the coordinates of its ce…
  7. 7A circle with centre O passes through the points P(-1,7) and Q(0,8). State the relationship between the perpendicular bi…
  8. 8Two points A(−3,−6)A(-3, -6) and B(7,−6)B(7, -6) lie on a circle with centre GG. The equation of the tangent to the circle at the…
  9. 9圆CC的半径为55，圆心在xx轴的负半轴上，且被直线3x+4y+4=03x+4y+4=0截得的弦长为66，则圆CC的方程为(      )\left(\;\;\;\right) A. x2+y2−2x−3=0x^{2}+y^{2}-2x-3=0…
  10. 10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…
  11. 11求过A(1,0)A(1,0)与B(0,1)B(0,1)两点，且在xx轴上截得的弦长等于66的圆的方程．
  12. 12A circle, C1C_1, has an equation x2−6x+y2+10y=66x^2 - 6x + y^2 + 10y = 66. Are the xx-axis and yy-axis tangents to C1C_1? Explain y…
  13. 13The equation of a circle C1C_1 is given by the equation x2+y2−4x−8y=0x^2 + y^2 - 4x - 8y = 0. Find the coordinates of the center an…
  14. 14已知圆的方程(x−2)2+y2=1{{(x-2)}^{2}}+{{y}^{2}}=1，过圆外一点P(3,4)P\left(3,4\right) 作一条直线与圆交于A,BA,B两点，那么PA→⋅PB→=\overrightarrow{PA}\cdot \overrightarrow{PB}=…
  15. 15已知圆心为CC的圆过点A(0,−6)A(0,-6)和B(1,−5)B(1,-5)，且圆心在直线ll：x−y+1=0x-y+1=0上． (1)求圆心为CC的圆的标准方程； (2)过点M(2,8)M(2,8)作圆的切线，求切线方程．
  16. 16A(1, 7) and B(9, 7) lie on a circle C1C_1 with centre P. The line 4y+3x=314y + 3x = 31 is a normal to the circle and passes th…
  17. 17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…
  18. 18已知圆CC：(x+1)2+(y−1)2=1(x+1)^{2}+(y-1)^{2}=1与xx轴切于AA点，与yy轴切于BB点，设劣弧A^B \hat AB的中点为MM，则过点MM的圆CC的切线方程是()() A…
  19. 19从点P(3,−1)P(3,-1)向圆x2+y2−4y−5=0x^{2}+y^{2}-4y-5=0作切线PT(TPT(T为切点))，则∣PT∣|PT|等于()(\quad) A. 5                                           5\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: …
  20. 20Solutions to this question by accurate drawing will not be accepted. The diagram shows an isosceles triangle PQR in whic…
  21. 21The equation of a circle is x2+y2+2x−8y−8=0x^2 + y^2 + 2x - 8y - 8 = 0. Find the coordinates of the centre, CC and the radius of the…
  22. 22The points PP and QQ both lie on a circle and have coordinates (−10,15)(-10, 15) and (10,0)(10, 0) respectively. The centre of t…
  23. 23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}…
  24. 24已知三个点A(0,0)A(0,0)，B(2,0)B(2,0)，C(4,2)C(4,2)，则△ABC\triangle ABC的外心的纵坐标是()(\quad) A. 32\dfrac{3}{2} B. 33 C. 52\dfrac{5}{2}…
  25. 25The lines y=9y=9, y=−1y=-1 and 3y=4x+93y=4x+9 are tangents to a circle. The x-coordinate of the centre of the circle is positive…
  26. 26直线y=2xy=2x与圆C:(x−2)2+(y+1)2=14C:(x-2{)}^{2}+(y+1{)}^{2}=14交于AA，BB两点，则△ACB{\rm\triangle}ACB的面积等于().(\quad). A. 352\dfrac{3\sqrt[]{5}}{2}…
  27. 27在极坐标系中，圆心为(1,π2)(1, \dfrac {π}{2})，且过极点的圆的方程是()() A. ρ=2sin⁡θρ=2\sin θ B. ρ=−2sin⁡θρ=-2\sin θ C. ρ=2cos⁡θρ=2\cos θ D. ρ=−2cos⁡θρ=-2\cos θ
  28. 28The equation of a circle, C1C_1, is Find the coordinates of the centre of C1C_1 and its radius. Find the equation of the…
  29. 29如果方程x2+y2+x+y+k=0x^{2}+y^{2}+x+y+k=0表示一个圆，则kk的取值范围是()(\quad) A. k>12k>\dfrac{1}{2} B. 0<k<120< k< \dfrac {1}{2} C. k<12k< \dfrac {1}{2}…
  30. 30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…
  31. 31圆x2+y2−mx+y+m=0x^{2}+y^{2}-mx+y+m=0在xx轴上截得的弦长是它在yy轴上截得的弦长的22倍，则实数mm的值是()(\quad) A. −6−210-6-2\sqrt{10} B. −6+210-6+2\sqrt{10}…
  32. 32过三点A(0,0)A(0,0)，B(0,2)B(0,2)，C(2,0)C(2,0)的圆MM与直线ll：kx−y+2−2k=0kx-y+2-2k=0的位置关系是()(\quad) A. 相交 B. 相切 C. 相交或相切 D. 相切或相离
  33. 33直线ll：kx−y+1=0kx-y+1=0被圆x2+y2−4y=0x^{2}+y^{2}-4y=0截得的最短弦长为()() A. 232 \sqrt {3} B. 33 C. 222 \sqrt {2} D. 22
  34. 34已知OO为坐标原点，M(1,2)M(1,2)，PP是抛物线CC：y2=2px(p>0)y^{2}=2px(p>0)上的一点，FF为其焦点，若点FF与双曲线x23−y2=1\dfrac{{{x}^{2}}}{3}-{{y}^{2}}=1的…
  35. 35已知圆方程C1C_{1}：f(x,y)=0f(x,y)=0，点P1(x1,y1)P_{1}(x_{1},y_{1})在圆C1C_{1}上，点P2(x2,y2)P_{2}(x_{2},y_{2})不在圆C1C_{1}上，则方程：f(x,y)−f(x1,y1)−f(x2,y2)=0f(x,y)-f(x_{1},y_{1})-f(x_{2},y_{2})=0…
  36. 36A circle, with centre CC, has equation x2+y2−10x−4y+25=0x^2 + y^2 -10x -4y +25 = 0. Find the coordinates of CC and the radius of the…
  37. 37A circle, C1C_1, has equation x2+y2−4x+6y=36x^2 + y^2 - 4x + 6y = 36. Find the radius and the coordinates of the centre of C1C_1.
  38. 38方程x2+y2+2x−4y−6=0x^{2}+y^{2}+2x-4y-6=0表示的图形是()() A. 以(1,−2)(1,-2)为圆心，1111为半径的圆 B. 以(−1,2)(-1,2)为圆心，1111为半径的圆 C. 以(−1,2)(-1,2)为圆心，11 \sqrt {11}…
  39. 39已知圆C1C_{1}：x2+y2=r2(r>0)x^{2}+y^{2}=r^{2}(r > 0)与抛物线C2C_{2}：x2=2py(p>0)x^{2}=2py(p > 0)，点(2,−2)( \sqrt {2},-2)是圆C1C_{1}与抛物线C2C_{2}…
  40. 40在直角坐标系xOyxOy中，曲线C1{{C}_{1}}的方程为y=k∣x∣+2.y=k\left| x \right|+2.以坐标原点为极点，xx轴正半轴为极轴建立极坐标系，曲线C2{{C}_{2}}的极坐标方程为ρ2+2ρcos⁡θ−3=0{{\rho }^{2}}+2\rho \cos \theta -3=0…
  41. 41The diagram shows part of the curve y=2x+4y = \sqrt{2x + 4}, meeting the xx-axis at the point AA and the yy-axis at the p…
  42. 42The diagram (not drawn to scale) shows a right-angled triangle PQR in which point P is (5, 4) and point R is (9, 2). Poi…
  43. 43Point PP lies on the curve y=kx22−xy = \frac{kx^2}{\sqrt{2-x}}, where kk is a constant. The xx-coordinate of PP is −2-2. G…
  44. 44The diagram shows a trapezium with vertices A(6, 8), B, C and D. The sides AB and DC are parallel and angle ABC is 90∘90^\circ…
  45. 45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}…
  46. 46In the diagram below, the line OPOP makes an angle of α∘\alpha^\circ with the positive xx-axis such that tan⁡α=0.2\tan \alpha = 0.2…
  47. 47Solutions to this question by accurate drawing will not be accepted. OABCOABC is a trapezium with right angle OABOAB. T…
  48. 48The points A(5,7)A(5, 7) and B(−11,15)B(-11, 15) lie on a circle CC. The centre of the circle OO lies on the line y=−2x−3y = -2x - 3.…
  49. 49已知直线ll：kx+y−2=0(k∈R)kx+y-2=0(k∈R)是圆CC：x2+y2−6x+2y+6=0x ^{2} +y ^{2} -6x+2y+6=0的一条对称轴，过点A(0,k)A(0 , k)作圆CC的一条切线，切点为BB，则线段ABAB的长度为…
  50. 50圆x2+y2−2x−5=0x^{2}+y^{2}-2x-5=0和圆x2+y2+2x−4y−4=0x^{2}+y^{2}+2x-4y-4=0的交点为AA、BB，则线段ABAB的垂直平分线方程为()(\quad) A. x+y−1=0                           x+y-1=0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: …
  51. 51The equation of a circle, C1C_1 is x2+y2−4x+6y−3=0x^2 + y^2 - 4x + 6y - 3 = 0. Find the coordinates of the centre and the radius of C1C_1…
  52. 52已知圆CC的圆心在直线y=x+1y=x+1上，且圆CC与xx轴相切，点P(−5,−2)P(-5,-2)在圆CC上，点Q(−4,−5)Q(-4,-5)在圆CC外． (1)求圆CC的方程； (2)若过点(−2,−4)(-2,-4)的…
  53. 53直线ll：mx+y−m−2=0mx+y-m-2=0与圆CC：(x−3)2+(y−4)2=25(x-3)^{2}+(y-4)^{2}=25交于AA，BB两点，CC为圆心，当∠ACB∠ACB最小时，直线ll的方程是 ___ ．
  54. 54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…
  55. 55求圆 x2+y2−4x+6y−12=0x^2 + y^2 - 4x + 6y - 12 = 0 的圆心坐标和半径。
  56. 56It is given that f(x)={4a2−x2for 0≤x≤2a,3x−6afor 2a<x<3a,f(x) = \begin{cases} \sqrt{4a^2 - x^2} & \text{for } 0 \le x \le 2a, \\ 3x - 6a & \text{for } 2a < x < 3a, \end{cases}…
  57. 57The diagram below is not drawn to scale. In the diagram, D, E and F are points on the circle C1. Show that DE is the dia…
  58. 58The equation of a circle with centre C is x2+y2−8x−2y−152=0x^{2}+y^{2}-8x-2y-152=0. Find the coordinates of C and the radius of the cir…
  59. 59A tangent to a circle at the point (−3,3)(-3,3) passes through the origin. Find the equation of the normal to the circle at…
  60. 60A circle, C1C_1, has the equation x2+y2+2x−6y=10x^2 + y^2 + 2x - 6y = 10. Find the coordinates of the centre and the exact radius of…