以点 A(1,2) 与 B(3,4) 为直径端点的圆的标准方程为＿＿＿＿＿＿。 (x-2) 2 +(y-3) 2 =2 圆的方程圆的方程直径中点坐标全国甲卷 2024

( x − 2 ) 2 + ( y − 3 ) 2 = 2 圆心为直径的中点： ( 2 1 + 3 , 2 2 + 4 ) = ( 2 , 3 ) 。半径为直径的一半： ∣ A B ∣ = ( 3 − 1 ) 2 + ( 4 − 2 ) 2 <path d="M263,681c0.7,0,18,39.7,52,119 c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 c340,-704.7,510.7,-1060.3,512,-1067 l0 -0 c4.7,-7.3,11,-11,19,-11 H40000v40H1012.3 s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232 c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z M1001 80h400000v40h-400000z"/> = 2 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/> ，故 r = 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/> 。代入标准方程： ( x − 2 ) 2 + ( y − 3 ) 2 = 2 。

高考数学考点题库/直线与圆/圆的方程

2024 年全国甲卷·第12题

填空题★★★☆☆ 中档改编题通法

以点

A (1, 2)

与

B (3, 4)

为直径端点的圆的标准方程为＿＿＿＿＿＿。

答案

(x - 2)^{2} + (y - 3)^{2} = 2

分步解析

1圆心为直径的中点： $(\frac{1 + 3}{2}, \frac{2 + 4}{2}) = (2, 3)$ 。
2半径为直径的一半： $∣ A B ∣ = (3 - 1)^{2} + (4 - 2)^{2} = 22$ ，故 $r = 2$ 。
3代入标准方程： $(x - 2)^{2} + (y - 3)^{2} = 2$ 。

#圆的方程#直径#中点坐标

2024 年真题全国甲卷

同考点推荐直线与圆 · 圆的方程

第8题单选★★☆☆☆直线与圆·圆的方程#圆的方程 #配方 #圆心半径改编查看 →第9题单选★★★☆☆直线与圆·弦长与切线#弦长公式 #点到直线距离 #圆改编查看 →第14题填空★★★★☆直线与圆·弦长与切线#切线长 #勾股定理 #圆改编查看 →第14题填空★★★☆☆直线与圆·直线与圆的位置关系#相切 #圆心距 #参数求值改编查看 →

想按考点连续练习？前往直线与圆专题页查看全部真题与解析。