Geometric Construction and Equation of a Parabola
2021-01-07 02:30
This example demonstrates the geometric construction and equation of a parabola. Using commands like Mirror, OrthogonalLine, and Locus, it dynamically illustrates the parabola as the locus of points equidistant from a focus and directrix. It helps learners intuitively understand its definition, axis of symmetry, and standard equation through interactive coordinate geometry.
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设抛物线的对称轴是x-y-2=0, 顶点是(4,2),\\
焦点是(2,0)，求该抛物线的方程。""/>
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到焦点和到准线的距离相等。\\
先求出准线方程。\\
1-1 准线垂直于对称轴，且垂足是焦点关于顶点的对称点。\\
即 2T_p=F_p+P, P=2T_p-F_p=2(4,2)-(2,0)=(6,4)\\
1-2 对称轴的斜率 k_1=1, 由准线\perp对称轴，斜率之积为-1\\
故准线斜率 k_2=\dfrac{-1}{k_1}=-1\\
1-3 由点斜式可得 准线方程为 y=4-1\times(x-6),化成 \\
x+y-10=0\\ \\

2. 由抛物线几何定义，设抛物线上任意点M(x,y)\\
|MF_p|=Distance(M,准线)\\
(x-2)^2+(y-0)^2=\dfrac{(x+y-10)^2}{2}, 展开化简\\
x^2-2xy+y^2+12x+20y-92=0""/>
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GGB https://www.geogebra.org/u/kumath""/>
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