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・圆的方程与切线挑战
A circle has center O, and points A and B lie on the circle. The measure of arc AB is 45∘ and the length of arc AB is 3 inches. What is the circumference, in inches, of the circle?
原库参考解析
圆・圆的方程与切线挑战
(x -6)2 + (y +5)2 = 16
In the xy-plane, the graph of the equation above is a circle. Point P is on the circle and has coordinates \( \left(10,-5\right) \) . If \( PQ \) is a diameter of the circle, what are the coordinates of point Q ?
原库参考解析
圆・圆的方程与切线挑战
A circle in the xy-plane has its center at (-5, 2) and has a radius of 9. An equation of this circle is x2 + y2 + ax + by + c = 0, where a, b, and c are constants. What is the value of c ?
原库参考解析
圆・圆的方程与切线挑战
The equation x 2 + (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
Points A and B lie on a circle with radius 1, and arc \( AB \) has length \( \frac{π}{3} \) . What fraction of the circumference of the circle is the length of arc \( AB \)?
原库参考解析
圆・圆的方程与切线挑战
The circle above has center O, the length of arc
is 5π, and x = 100 . What is the length of arc 
?原库参考解析
圆・圆的方程与切线挑战
A circle in the xy-plane has equation (x +3)2 + (y -1)2 = 25. Which of the following points does NOT lie in the interior of the circle?
原库参考解析
圆・圆的方程与切线挑战
The graph of x2 + x + y2 + y = \( \frac{199}{2} \)in the xy-plane is a circle. What is the length of the circle’s radius?
原库参考解析
圆・圆的方程与切线挑战
x2 + 20x + y2 + 16y = -20
The equation above defines a circle in the xy-plane. What are the coordinates of the center of the circle?
原库参考解析
圆・圆的方程与切线挑战
Point O is the center of the circle above, and the measure of ∠OAB is 30°. If the length of \( OC \) is 18, what is the length of arc \( AB \) ?
原库参考解析
圆・圆的方程与切线挑战
A circle in the xy-plane has a diameter with endpoints 2, 4 and 2, 14. An equation of this circle is \( \left(x-2\right)^2+\left(y-9\right)^2=r^2 \) , where r is a positive constant. What is the value of r ?
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
