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(-4, 2) and the point(-8, 5) lies on the circle. Which equation represents this circle?
原库参考解析
圆・圆的方程与切线进阶
\( x^2+9x+y^2-5y+17=0 \)
In the xy-plane, the graph of the given equation is a circle. 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 the equation\( \left(x-14\right)^2+\left(y-k\right)^2=36 \). Which of the following gives the center of the circle and its radius?
原库参考解析
圆・圆的方程与切线进阶
In the xy-plane, circle M is the graph of the equation \( \left(x-3\right)^2+\left(y-6\right)^2=4 \) . Circle P has the same center as circle M but has a radius that is twice the radius of circle M. Which equation represents circle P?
原库参考解析
圆・圆的方程与切线挑战
A rectangle is inscribed in a circle such that the length of the diagonal of the rectangle is twice the length of its shortest side. The circumference of the circle is 114π units. What is the area, in square units, of the rectangle?
原库参考解析
圆・圆的方程与切线挑战
The three points shown define a circle. The circumference of this circle is kπ, 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
\( \left(x-5\right)^2+\left(y+3\right)^2=36 \)
The given equation represents circle P in the xy-plane. Circle Q has a center that is 3 units to the right of and 2 units below the center of circle P. Circle Q has a diameter that is double the diameter of circle P. Which equation represents circle Q?
原库参考解析
圆・圆的方程与切线挑战
Circle \( F \)in the xy-plane is represented by the equation \( \left(x-13\right)^2+\left(y-9\right)^2=196 \). Circle G is obtained by shifting circle \( F 5 \) units to the left and 11 units up. An equation representing circle \( G \) is \( \left(x+h\right)^2+\left(y+k\right)^2=196 \), where h and k are constants. What is the value of \( h+k \) ?
原库参考解析
圆・圆的方程与切线进阶
\( (x−3)^{2}+(y−9)^{2}=6 \)
In the xy-plane, the graph of the given equation is a circle. Which point lies on this circle?
原库参考解析
圆・圆的方程与切线挑战
In the figure shown, points A, B, C, and E lie on the circle, and AB < BC . Segment AC is perpendicular to segment BE at point D, and BD =\( \sqrt{390} \) .The diameter of the circle is 197. If\( \frac{CD}{AD}=r \), what is the value of r ?
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・圆的方程与切线进阶
In the xy-plane, the graph of the equation \( \left(x-4\right)^2+\left(y-2\right)^2=16 \) is a circle. The point \( \left(8,\ c\right) \), where c is a constant, lies on this circle. What is the value of c?
原库参考解析
