Interactive Conic Sections Problem-Solving Demo
2020-02-29 06:53
This GeoGebra example demonstrates the geometric construction process for solving conic section problems. Using commands such as Intersect, OrthogonalLine, Ray, Segment, and Line, it builds geometric figures and auxiliary lines related to conic sections. The applet features multiple key points (A, B, C, D, E, etc.) and line objects, along with InputBox elements for interactive parameter entry, dyn
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<expression label="text1" exp=""由已知，直线AM的方程为：y=kx+m\\ 
∴N(-\frac{m}{k},0)\\
∵M是AN的中点\\
∴A(\frac{m}{k},2m)\\
∴D(\frac{m}{k},-2m)\\
∴k^{'}=\frac{3m}{-\frac{m}{k}}=-3k\\
∴3k+k^{'}=3k+(-3k)=0，即证""/>
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<expression label="text2" exp=""直线AM的方程联立椭圆C的方程，得\\
\begin{cases} y=kx+m， \\ \frac{x^2}{6}+\frac{y^2}{3}=1， \end{cases} 消去y,得(1+2k^2)x^2+4kmx+2m^2-6=0\\
利用韦达定理可知：x_{A}·x_{B}=\frac{2m^2-6}{1+2k^2},∴x_{B}=\frac{2m^2-6}{(1+2k^2)x_{A}}\\
∵3k+k^{'}=0,x_{A}=x_{D}\\
∴同理可得，x_{G}=\frac{2m^2-6}{(1+2k^{'}^2)x_{A}}=\frac{2m^2-6}{(1+18k^2)x_{A}}\\
∴k_{BG}=\frac{y_{B}-y_{G}}{x_{B}-x_{G}}=\frac{kx_{B}+m-(-3kx_{G}+m)}{x_{B}-x_{G}}=\frac{kx_{B}+3kx_{G}}{x_{B}-x_{G}}\\
=k\frac{\frac{2m^2-6}{(1+2k^2)x_{A}}+3\frac{2m^2-6}{(1+18k^2)x_{A}}}{\frac{2m^2-6}{(1+2k^2)x_{A}}-\frac{2m^2-6}{(1+18k^2)x_{A}}}\\
=\frac{1+6k^2}{4k}=\frac{1}{4k}+\frac{3k}{2}\\
∵k>0,k_{BG}=\frac{1}{4k}+\frac{3k}{2}≥2\sqrt{\frac{1}{4k}·\frac{3k}{2}}=\frac{\sqrt{6}}{2}\\
当且仅当\frac{1}{4k}=\frac{3k}{2}时，即k=\frac{\sqrt{6}}{6}时，等号成立\\
故直线BG斜率的最小值为\frac{\sqrt{6}}{2}。\\""/>
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