圆锥曲线的弦长公式推导 直线截椭圆的弦长公式,要详细证明,一步步推导~谢谢~!
\u6c42\u89e3\u5706\u9525\u66f2\u7ebf\u7684\u5f26\u957f\u516c\u5f0f\u7684\u63a8\u5bfc\u8fc7\u7a0by=kx+b
\u5f26\u957fd=\u221a[(x1-x2)^2+(y1-y2)^2]
=\u221a[(x1-x2)^2+k^2(x1-x2)^2]
=\u221a\uff081+k^2\uff09\u221a[(x1-x2)^2]
=\u221a\uff081+k^2\uff09\u221a[(x1+x2)^2-
4x1x2]
\u5982\u679c\u7528y\u6765\u8868\u793a
x=1/k(y-b)
\u5c31\u4f1a\u5f97\u5230d
=
\u221a\uff081+k^2\uff09[(x1+x2\uff09^2
-
4x1x2]
=
\u221a\uff081+1/k^2\uff09[(y1+y2\uff09^2
-
4y1y2]
弦长d=√[(x1-x2)^2+(y1-y2)^2]
=√[(x1-x2)^2+k^2(x1-x2)^2]
=√(1+k^2)√[(x1-x2)^2]
=√(1+k^2)√[(x1+x2)^2- 4x1x2]
如果用y来表示
x=1/k(y-b)
就会得到d = √(1+k^2)[(x1+x2)^2 - 4x1x2] = √(1+1/k^2)[(y1+y2)^2 - 4y1y2]
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