过点 P(3,4) 作圆 x 2 +y 2 =9 的切线，则切线长为＿＿＿＿＿＿。 4 弦长与切线切线长勾股定理圆北京卷 2023

4 切线长、半径、圆心到外点的距离构成直角三角形。 ∣ O P ∣ = 3 2 + 4 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"/> = 5 ，半径 r = 3 。由勾股定理，切线长 = ∣ O P ∣ 2 − r 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"/> = 25 − 9 <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"/> = 4 。

高考数学考点题库/直线与圆/弦长与切线

2023 年北京卷·第14题

填空题★★★★☆ 较难改编题巧法

过点

P (3, 4)

作圆

x^{2} + y^{2} = 9

的切线，则切线长为＿＿＿＿＿＿。

答案

4

分步解析

1切线长、半径、圆心到外点的距离构成直角三角形。
2 $∣ O P ∣ = 3^{2} + 4^{2} = 5$ ，半径 $r = 3$ 。
3由勾股定理，切线长 $= ∣ O P ∣^{2} - r^{2} = 25 - 9 = 4$ 。

点评切线长公式

d^{2} - r^{2}

是圆的高频结论，其中

d

是外点到圆心的距离。

#切线长#勾股定理#圆

2023 年真题北京卷

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