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 \( (-7,\ 2) \), and the point \( (1,−2) \) lies on the circle. Which equation represents this circle?
原库参考解析
圆・圆的方程与切线挑战
\( x^2+y^2-6x-10y-4n=0 \)
The given equation represents circle A in the xy-plane, where n is a constant. Point (5, 7) lies on circle B, which has the same center but twice the diameter as circle A. What is the value of n?
原库参考解析
圆・圆的方程与切线挑战
Points A, B, and C lie on the circle as shown. If \( x=48 \), what is the measure, in degrees, of minor arc AC?
原库参考解析
圆・圆的方程与切线进阶
A circle in the xy-plane has its center at (3, 2) and a radius with a length of 4. The equation for this circle can be written in the form \( (x-h)^2+(y-k)^2=b \), where h, k, and b are constants. What is the value of h?
参考答案
圆・圆的方程与切线进阶
A circle has a circumference of 28π centimeters. What is the diameter, in centimeters, of the circle?
原库参考解析
圆・圆的方程与切线进阶
In the xy-plane, an equation of circle A is \( (x-2)^2+(y-7)^2=25 \) . Circle B has the same center as circle A but has a radius that is twice the radius of circle A. Which equation represents circle B?
原库参考解析
圆・圆的方程与切线进阶
The circle shown has its center at (0,0). What is the value of k?
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 247 millimeters and the perimeter of quadrilateral GMHN is 5,174 millimeters, what is the distance, in millimeters, between points G and H?
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, 5). Line t is tangent to this circle at the point (6, -1). Which of the following points also lies on line t ?
原库参考解析
圆・圆的方程与切线进阶
A circle in the xy-plane has a diameter with endpoints (8, -13) and (8, -25). Which equation represents this circle?
原库参考解析
圆・圆的方程与切线挑战
\( 2x^2+6x+2y^2+20y=\frac{11}{2} \)
The given equation defines a circle in the xy-plane. What is the radius of the circle?
原库参考解析
圆・圆的方程与切线进阶
A circle in the xy-plane has its center at (6, 5) and has a radius of length 8. Which equation represents this circle?
原库参考解析
