这部分需要会什么？
- 由标准式读取圆心和半径
- 用配方把一般式改成圆的标准式
- 使用半径与切线垂直及圆周角关系
核心概念与公式
圆的标准式为 \((x-h)^2+(y-k)^2=r^2\)，圆心 \((h,k)\)，半径 \(r>0\)。一般式中若 \(x^2,y^2\) 系数相同且非零，可以先归一化，再分别配方；右边得到正数才表示非退化的实圆。
圆的切线在切点处垂直于半径。从圆外同一点引出的两条切线段长度相等。同弧所对的圆周角为圆心角的一半；直径所对的圆周角为直角。
做题时怎样使用
- 有方程先化标准式，注意圆心坐标的符号和半径平方
- 有切线先连接圆心与切点，寻找直角三角形
- 有角和弧时明确对应的是同一条弧，再使用角度关系
容易混淆的地方
标准式右边是 \(r^2\)，不是 \(r\)。\((x+2)^2\) 表示圆心横坐标 \(-2\)。单看直线经过圆上一点不能确定它是切线。
用一个例题串起方法
知识点教学例题求 \(x^2+y^2-6x+4y-12=0\) 的圆心与半径。
展开解题思路 ＋
配方得 \((x-3)^2+(y+2)^2=12+9+4=25\)。因此圆心为 \((3,-2)\)，半径为 5。
本知识点题库
先独立作答，再查看答案与解析。
圆・圆的方程与切线进阶
A circle in the xy-plane has its center at (19,18) and has a radius of \( 9k \). Which equation represents this circle?
原库参考解析
圆・圆的方程与切线基础
Circle N has a radius of 7 millimeters (mm). Circle M has an area of 64π mm². What is the total area, in mm², of circles N and M?
原库参考解析
圆・圆的方程与切线挑战
Circle A in the xy-plane has the equation \( (x+5)^2+(y-5)^2=25 \). Circle B has the same center as circle A. The radius of circle B is two times the radius of circle A. The equation defining circle B in the xy-plane is \( (x+5)^2+(y-5)^2=k \), where k is a constant. What is the value of k?
原库参考解析
圆・圆的方程与切线进阶
The circle shown has center O, circumference 144π, and diameters PR and QS. The length of arc PS is twice the length of arc PQ.What is the length of arc QR
原库参考解析
圆・圆的方程与切线挑战
Circle A in the xy-plane has the equation (x ＋5)2+(y -5)2= 4.Circle B has the same center as circle A. The radius of circle B is two times the radius of circle A.The equation defining circle B in the xy-plane is (x＋5)2+(y- 5)2= k, where k is a constant. What is the value of k?
原库参考解析
圆・圆的方程与切线进阶
Point O is the center of a circle. The measure of arc RS on this circle is 100°. What is the measure, in degrees, of its associated angle ROS?
原库参考解析
圆・圆的方程与切线基础
The equation x2 + (y - 2)2 = 36 represents circle A. Circle B is obtained by shifting circle A down 4 units in the xy-plane. Which of the following equations represents circle B?
原库参考解析
圆・圆的方程与切线挑战
The graph of x2 + x + y2 + y =
原库参考解析
圆・圆的方程与切线进阶
\( \left(x+4\right)^2+\left(y-19\right)^2=121 \)
The graph of the given equation is a circle in the \( xy \)-plane. The point (a, b) lies on the circle. Which of the following is a possible value for a?
原库参考解析
圆・圆的方程与切线挑战
In the xy-plane, a circle has center C with coordinates (h, k). Points A and B lie on the circle. Point A has coordinates \( (h+1,k+\surd 102), \)and ∠ACB is a right angle. what is the length of \( AB \) ?
原库参考解析
圆・圆的方程与切线进阶
Circle A (shown) is defined by the equation \( \left(x+2\right)^2+y^2=9 \) . Circle B (not shown) is the result of shifting circle A down 6 units and increasing the radius so that the radius of circle B is 2 times the radius of circle A. Which equation defines circle B?
原库参考解析
圆・圆的方程与切线挑战
A circle has center G, and points M and N lie on the circle. Line segments MH and NH are tangent to the circle at points M and N, respectively, If the radius of the circle is 168 millimeters and the perimeter of quadrilateral GMHN is 3,856 millimeters, what is the distance, in millimeters, between points G and H?
原库参考解析
