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
\( 2x^2+18x+2y^2+16y=\frac{11}{2} \)
The given equation defines a circle in the xy-plane. What is the radius of the circle?
原库参考解析
圆・圆的方程与切线挑战
A circle has center P, and points A and B lie on the circle. The measure of arc AB is 30° and the length of arc AB is 7π units. What is the length, in units, of the radius of the circle?
参考答案
圆・圆的方程与切线进阶
In the circle, O is the center and \( ∠AOB \) is a right angle. If the area of the shaded region is 37π square units, what is the length, in units, of \( OA \)?
参考答案
圆・圆的方程与切线挑战
\( x^2+y^2=49 \)
\( y=mx+\frac{b}{3} \)
In the given system of equations, m and b are negative constants. In the xy-plane, the graphs of the equations in the given system intersect at the point (−4, y) where y < 0. Which expression represents the value of b?
参考答案
圆・圆的方程与切线进阶
The circle shown has center Q, and points B, C, and D lie on the circle. Line m is tangent to the circle at point C, and line m is parallel to line segment BD(not shown). Line DQ(not shown) intersects line m at point A, and line BQ(not shown) intersects line m at point P. If the length of line segment AQ(not shown) is 90 and the length of line segment AC is 72, what is the perimeter of triangle CPQ?
参考答案
圆・圆的方程与切线进阶
Points A and B lie on a circle with center C. The length of segment AC is 86 millimeters. What is the length, in millimeters, of segment BC?
参考答案
圆・圆的方程与切线挑战
\( x^2+16x+y^2=12y+101 \)
In the xy-plane, the graph of the given equation is a circle. What is the length of the circle's diameter?
In the xy-plane, the graph of the given equation is a circle. What is the length of the circle's diameter?
参考答案
圆・圆的方程与切线挑战
A circle in the xy-plane has its center at (-5, 2) and the point (-9, 5) lies on the circle. Which equation represents this circle?
参考答案
圆・圆的方程与切线挑战
A circle in the xy-plane has its center at (-2, 3) and has a radius of 8. An equation of this circle is \( x^2+y^2+ax+by+c=0 \), where a, b, and c are constants. What is the value of c?
参考答案
圆・圆的方程与切线进阶
\( x^2+y^2-8x+4y-29=0 \)
In the xy-plane, the graph of the given equation is a circle. What are the coordinates (x, y) of the center of the circle?
In the xy-plane, the graph of the given equation is a circle. What are the coordinates (x, y) of the center of the circle?
参考答案
圆・圆的方程与切线挑战
\( x^2-16x+y^2+20y=32 \)
In the xy-plane, the graph of the given equation is a circle. What is the length of the circle’s diameter?
In the xy-plane, the graph of the given equation is a circle. What is the length of the circle’s diameter?
参考答案
圆・圆的方程与切线挑战
\( x^2-6x+y^2+20y=12 \)
In the xy-plane, the graph of the given equation is a circle. What is the length of the circle’s diameter?
In the xy-plane, the graph of the given equation is a circle. What is the length of the circle’s diameter?
参考答案
