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・圆的方程与切线基础
In the xy-plane, segment QR is a diameter of circle O. Line s and line t are tangent to circle O at points Q and R, respectively. The slope of line s is \( \frac{3}{2} \). What is the slope of line t?
参考答案
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?
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
A circle in the xy-plane has its center at \( (-4,\ -8) \). Line k is tangent to this circle at the point \( (-8,\ -9) \). What is the slope of line k?
参考答案
圆・圆的方程与切线基础
A circle has a diameter of 10.4 centimeters. What is the radius of the circle, in centimeters?
参考答案
圆・圆的方程与切线挑战
\( x^2+x\sqrt{t}+y^2-y-73=0 \)
The given equation, where t is a constant, defines a circle in the xy-plane. The radius of this circle is \( \sqrt{91} \). What is the value of t?
The given equation, where t is a constant, defines a circle in the xy-plane. The radius of this circle is \( \sqrt{91} \). What is the value of t?
参考答案
圆・圆的方程与切线进阶
\( \left(x+3\right)^2+\left(y-18\right)^2=169 \)
The graph of the given equation is a circle in the xy-plane. The point (a, b) lies on the circle. Which of the following is a possible value for a?
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 \( \left(x-2\right)^2+\left(y-9\right)^2=4 \) is a circle. The point (4, c), where c is a constant, lies on this circle. What is the value of c?
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
\( x^2+y^2-10x-8y-14n=0 \)
The given equation represents circle A in the xy-plane, where n is a constant. Point (11, 8) 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?
原库参考解析
圆・圆的方程与切线挑战
Circle A has a radius of 3n and circle B has a radius of 129n, where n is a positive constant. The area of circle B is how
many times the area of circle A?
原库参考解析
圆・圆的方程与切线挑战
Which of the following equations represents a circle in the xy-plane that intersects the y-axis at exactly one point?
原库参考解析
