Parabola

One of the conic sections. The locus of points equidistant from a fixed line l and a fixed point F is a parabola, where F is called the focus and l is the directrix ( Figure A ). A parabola is symmetrical with respect to a line that passes through F and is perpendicular to l. This line is called the parabola's (principal) axis, and the intersection of the parabola and the axis is called the vertex. In a Cartesian coordinate system with the vertex O as the origin and the x-axis as the axis, the equation for a parabola is of the form y ² = 4px, with x = -p being the directrix. A parabola is the curve described by an object thrown up at an angle as it falls. Light entering parallel to the principal axis is reflected by the parabola's mirror and all comes to a focus ( Figure B ).

^y2 = 4k(x + k) (-∞ < k < ∞)
is a single parabola for each value of k, but regardless of the value of k, they all have the same focus (0,0) and the same axis y=0. In addition, there are parabolas that pass through any point other than the origin and correspond to exactly two values ​​of k, and they intersect at right angles at that point. In this case, the point (0,0) is called the confocal point of the group of parabolas ( Figure C ).

[Tachibana Shunichi]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

円錐曲線(えんすいきょくせん)の一つ。一定直線lと一定点Fとから等距離にある点の軌跡が放物線であり、Fを焦点、lを準線という（図A）。Fを通りlに垂直な直線に関して放物線は対称である。この直線を放物線の（主）軸、放物線と軸との交点を頂点という。頂点Oを原点とし軸をx軸とする直交座標系に関して、放物線の方程式はy²=4pxの形で、x=-pが準線である。放物線は、斜めに投げ上げられた物体が落下するときに描く曲線である。主軸に平行に入ってきた光は放物線の鏡で反射し、すべて焦点に集まる（図B）。

　　y²=4k(x+k) (-∞＜k＜∞)
は、kの値を一つ定めると一つの放物線であるが、kの値に関係なくすべて同一の焦点(0,0)と同一の軸y=0をもつ。また、原点以外の任意の点を通りちょうど二つのkの値に対応する放物線があって、それらはその点で互いに直交する。このとき点(0,0)を放物線群の共焦点という（図C）。

［立花俊一］

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