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 (14, 19) and has a radius of 7k. Which equation represents this circle?
原库参考解析
圆・圆的方程与切线进阶
In the xy-plane, circle F is defined by the equation \( \left(x+7\right)^2+\left(y+7\right)^2=36 \) . Circle G has the same center as circle F, and the radius of circle G is 2 units greater than the radius of circle F. Circle G is defined by the equation \( \left(x+7\right)^2+\left(y+7\right)^2=p \), where p is a constant. What is the value of p?
原库参考解析
圆・圆的方程与切线进阶
\( \left(x+5\right)^2+\left(y+8\right)^2=15^2 \)
In the xy-plane, the graph of the given equation is a circle. What is the length of a radius of this circle?
原库参考解析
圆・圆的方程与切线挑战
\( 2x^2+2y^2+tx+ty-78=0 \)
The given equation, where t is a positive constant, defines a circle in the xy-plane. The radius of this circle is \( \sqrt{41} \) ,what is the value of t?
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 \( \left(-1,\ 1\right) \) . Line t is tangent to this circle at the point \( \left(7,\ -6\right) \). 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
Circle A shown is defined by the equation \( x^2+\left(y-5\right)^2=7 \) . Circle B (not shown) has the same radius but is translated 91 units to the right. If the equation of circle B is \( \left(x-h\right)^2+\left(y-k\right)^2=a \) , where h, k, and a are constants, what is the value of \( 4a \) ?
原库参考解析
圆・圆的方程与切线进阶
A circle in the xy-plane has center \( (28,\ 5) \) and y-intercepts \( (0,−16) \) and \( (0,\ 26) \). What is the radius of this circle?
原库参考解析
圆・圆的方程与切线进阶
A circle has center O, and points A and B lie on the circle. The measure of arc AB is 60°, and the length of this arc is 4 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
A circle in the xy-plane has its center at \( (-7,\ 3) \) and has a radius of 9. 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 ?
原库参考解析
圆・圆的方程与切线进阶
Circles A and B are graphed in the xy-plane. Circle A is represented by the equation \( \left(x+8\right)^2+\left(y-8\right)^2=64 \) and intersects the x-axis at the point (r, s). Circle B has its center at (r, s) and has a radius of the same length as circle A. Which equation represents circle B?
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 \( (2,\ 9) \). Line t is tangent to this circle at the point \( (a,−4) \), where a is a constant. The slope of line t is \( \frac{6}{5} \). What is the value of a?
原库参考解析
圆・圆的方程与切线进阶
\( x^2+52x+y^2=0 \)
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?
原库参考解析
