Math topic中文

Geometric problems

geometric problems involving the use of coordinates

Practice questions

  1. 1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).).…
  2. 2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…
  3. 3What is the area of the triangle formed by the lines y=5y = 5, y=1+xy = 1 + x, and y=1−xy = 1 - x? A. 4 B. 8 C. 10 D. 12 E. 16
  4. 4若曲线y=x−12y=x^{\mathrm{{-}}\frac{1}{2}}在点(a,a−12)(a,a^{\mathrm{{-}}\frac{1}{2}})处的切线与两个坐标轴围成的三角形的面积为1818，则aa的值为___.
  5. 5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…
  6. 6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}…
  7. 7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
  8. 8A rectangle, with sides parallel to the xx-axis and yy-axis, has opposite vertices located at (15,3)(15, 3) and (16,5)(16, 5).…
  9. 9已知直线ll：3x+3y−1=03x+\sqrt{3}y-1=0，关于直线ll的以下四个结论： ①直线ll的倾斜角为2π3\dfrac{2\pi}{3}；②直线ll经过点(23,3)\left(\dfrac{2}{3},\sqrt{3}\right)…
  10. 10设函数f(x)=ax+bxf(x)=ax+ \dfrac {b}{x}，曲线y=f(x)y=f(x)在点M(3,f(3))M( \sqrt {3},f( \sqrt {3}))处的切线方程为2x−3y+23=02x-3y+2 \sqrt {3}=0． ((Ⅰ))求f(x)f(x)…
  11. 11在平面直角坐标系中，抛物线y1=x2−2x+a{y}_{1}={x}^{2}-2x+a 与xx轴的一个交点为AA，抛物线y2=x2+2a+1+2a{y}_{2}={x}^{2}+2a+1+2a 与xx轴的一个交点为BB，且AA、BB两…
  12. 12若圆x2+y2=r2(r>0)x^{2}+y^{2}=r^{2}(r>0)上有且仅有44个点到直线ll：x−y−2=0x-y-2=0的距离为11，则实数rr的取值范围是()(\quad) A. (2+1,+∞)(\sqrt{2}+1,+∞)…
  13. 13The coordinates of △ABC\triangle ABC are A(5,7)A(5, 7), B(11,7)B(11, 7) and C(3,y)C(3, y), with y>7y > 7. The area of △ABC\triangle ABC is…
  14. 14Given that the area of ΔACD\Delta ACD is 1.5 times that of ΔABC\Delta ABC, find the coordinates of BB. Showing your working…
  15. 15在极坐标系中，曲线CC：ρ=2acos⁡θ(a>0)ρ=2a\cos θ(a > 0)，直线ll：ρcos⁡(θ−π3)=32ρ\cos (θ- \dfrac {π}{3})= \dfrac {3}{2}，CC与ll有且只有一个公共点，求aa．
  16. 16已知直线l1l_{1}：4x−3y+6=04x-3y+6=0和直线l2l_{2}：x=−1x=-1，PP是抛物线y2=4xy^{2}=4x上一个动点，则点PP到直线l1l_{1}和直线l2l_{2}的距离之和的最小值是___…
  17. 17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…
  18. 18Line l passes through the points (-4, 4) and (8, -2). Find the equation of line l. Answer ... The equation of line m is…
  19. 19点A(−1,−2)A(-1,-2)关于直线l:x−y+1=0l:x-y+1=0的对称点AˈAˈ的坐标为 ___
  20. 20(1)已知线段aa、bb、cc满足aa：bb：c=3c=3：22：66，且a+2b+c=26a+2b+c=26． ①①求aa、bb、cc的值； ②②若线段xx是线段aa…