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
A circle in the xy-plane has its center at \( (2,\ 7) \). 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=DE=\( \sqrt{398} \). The diameter of the circle is 201. 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
In the xy-plane, an equation of circle A is \( (x-3)^2+(y-9)^2=25 \) . Circle B has the same center as circle A but has a radius that is twice the radius of 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 (-5, 2) and the point (-9, 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
A circle in the xy-plane has its center at (−4, 1) and the point (−8, 4) 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
What is the radius of the circle in the xy-plane defined by \( (x+3)^2+(y+9)^2=361 \) ?
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 45° and the length of arc AB is \( 4π \) units. What is the length, in units, of the radius of the circle?
原库参考解析
圆・圆的方程与切线挑战
\( x^2+y^2=36 \)
\( y=mx+\frac{b}{4} \)
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 \( (-5,\ y) \), where \( y<0 \). Which expression represents the value of b?
原库参考解析
圆・圆的方程与切线进阶
What is the radius of the circle in the xy-plane defined by \( (x+2)^2+(y+6)^2=361 \)?
原库参考解析
圆・圆的方程与切线进阶
\( x^2+y^2=1,088 \)
\( y-4x=0 \)
A solution to the given system of equations is \( (x,\ y) \), where \( x<0 \). What is the value of \( y \)?
原库参考解析
圆・圆的方程与切线挑战
Circle A: \( (x-7)^2+(y-p)^2=21 \)
Circle B: \( (x+7)^2+(y-p)^2=21 \)
In the given equations, p is a positive constant. Which statement correctly compares the graphs of circles A and B in the xy-plane?
原库参考解析
圆・圆的方程与切线进阶
In the xy-plane, an equation of circle R is \( (x+8)^2+(y+19)^2=100 \). Circle S is obtained by shifting circle R to the right 3 units. An equation defining circle S is \( (x+h)^2+(y+k)^2=100 \), where h and k are constants. What is the value of h?
原库参考解析
