Consider the set of parabolas having common chord of maximum length
Answers
what we have to prove..
Answer:
In mathematics, a parabola is a two-dimensional planar curve that is mirror-symmetric and shaped like the English letter U. A parabola is a two dimensional curve that is defined in many ways. One definition defines a parabola as a special form of a conic. According to this, a parabola is a conic whose eccentricity is equal to 1. A parabola can be defined as a locus. A parabola is a locus of points whose distance from a certain line is equal to their distance from a fixed point. Here, that line is called a directrix, and that point is called focus.
Step-by-step explanation:
For example, the equation x2 = 4ay represents a parabola with the direction y =-a and focus (0, a).
The line perpendicular to the coordinate, which passes through the focus (i.e., the line that bisects the parabola from the middle), is called the "symmetric axis" or "axis of the parabola". The point on the parabola, which intersects the axis—that is, the point of intersection of the axis of the parabola and the parabola—is called the "vertex of the parabola," and at this point, the parabola is most curved. The distance between the focus and the vertex, on the axis of the parabola, is called the "focal distance." The "focal" is the chord of the parabola that is parallel to the directrix (or perpendicular to the axis) and passes through the focus. The open side of the parabola can be up, down, left, right, or in any other desired direction. Any parabola can be restored and rescaled to fit exactly on another parabola, i.e. all parabolas are geometrically identical.
right open parabola
{\displaystyle y^{2}=4ax}{\displaystyle y^{2}=4ax}
Its vertex is the origin, the focus {\displaystyle (a,0)}{\displaystyle (a,0)} and the coordinate {\displaystyle x=-a}{\displaystyle x=-a} .
left open parabola
{\displaystyle y^{2}=-4ax}{\displaystyle y^{2}=-4ax}
Its vertex is the origin, the focus {\displaystyle (-a,0)}{\displaystyle (-a,0)} and the coordinate {\displaystyle x=a}{\displaystyle x=a} .
upward open parabola
{\displaystyle x^{2}=4ay}{\displaystyle x^{2}=4ay}
Its vertex is the origin, the focus {\displaystyle (0,a)}{\displaystyle (0,a)} and the coordinate {\displaystyle y=-a}{\displaystyle y=-a} .
downward open parabola
{\displaystyle x^{2}=-4ay}{\displaystyle x^{2}=-4ay}
Its vertex is the origin, the focus {\displaystyle (0,-a)}{\displaystyle (0,-a)} and the coordinate {\displaystyle y=a}{\displaystyle y=a} .