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
Circle A in the xy-plane has the equation \( \left(x+9\right)^2+\left(y-9\right)^2=4 \) . 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 \( \left(x+9\right)^2+\left(y-9\right)^2=k \) , 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
Points A, B, and C lie on a circle with a radius of 15 inches. If the length of arc ABC is 5π inches, what is the length, in inches, of chord AC ?
原库参考解析
圆・圆的方程与切线挑战
A circle has diameters \( AC \) and \( BD \) . The circumference of the circle is 84π, and the length of arc DA is 2 times the length of arc AB. What is the length of arc BC?
原库参考解析
圆・圆的方程与切线进阶
A circle is inscribed in a square such that the circumference of the circle touches the midpoint of each side of the square. The length of the diagonal of the square is 176 units. What is the area, in square units, of the circle?
原库参考解析
圆・圆的方程与切线挑战
x2 - 4x + y2 -12y -81 = 0
In the xy-plane, the graph of the given equation is a circle. If this circle is inscribed in a square, what is the perimeter of the square?
原库参考解析
圆・圆的方程与切线挑战
A circle has diameters \( AC \) and \( BD \) . The circumference of the circle is 168π, and the length of arc DA is 3 times the length of arc AB. What is the length of arc BC?
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 \( \left(x+8a\right)^2+\left(y-60a\right)^2=64a^2 \), where a is a positive constant. Circle V is obtained by shifting circle M 16a units to the right. Which equation represents circle V?
原库参考解析
圆・圆的方程与切线挑战
The perimeter of an equilateral triangle is 750 centimeters. The three vertices of the triangle lie on a circle. The radius of the circle is \( w\sqrt{3} \) centimeters. What is the value of w?
原库参考解析
圆・圆的方程与切线进阶
\( x^2-4x+y^2-8y-80=0 \)
In the xy-plane, the graph of the given equation is a circle. If this circle is inscribed in a square. What is the perimeter of the square?
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 (18, 16) and has a radius of 6k. Which equation represents this circle?
原库参考解析
圆・圆的方程与切线挑战
The perimeter of an equilateral triangle is 876 centimeters. The three vertices of the triangle lie on a circle. The radius of the circle is \( w\sqrt{3} \) centimeters. What is the value of w?
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 \( \left(-2,\ 2\right) \) . Line t is tangent to this circle at the point \( \left(3,-1\right) \). Which of the following points also lies on line t?
原库参考解析
