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Parabola

ஒரு பராபோலா என்பது ஒரு நிதியின், குறிக்கப்பட்ட ஒரு புள்ளியின்று, முனையாக, அதற்குதானே காட்சியளிப்பட்ட ஓர் வரியிடத்தில் உள்ள அனைத்து புள்ளிகளும் உள்ளவை.

Important concepts:

Standard form

• Standard form of a horizontal parabola: $$x=a{y}^2+by+c$$; if $$a<0$$ then the parabola opens to theleft; if $$a>0$$ then the parabola opens to the right.
• Standard form of a vertical parabola: $$y=a{x}^2+bx+c$$; if $$a<0$$ then the parabola opens downward like a frown; if $$a>0$$ then the parabola opens upward like a smile.

Vertex form
The vertex of a parabola is usually represented by $$h$$ (for the x-coordinate) and $$k$$ (for the y-coordinate), which can be found using the vertex form. In the vertex form for both horizontal and vertical parabolas, $$a$$ represents a reflection across the x-axis and/or a vertical stretch or compression, $$h$$ represents a horizontal translation (a shift to the left or right), and $$k$$ represents a vertical translation (a shift up or down).

• Vertex form of a horizontal parabola: $$x=a{\left(y-k\right)}^2+h$$, in which ; if $$a<0$$ then the vertex is on the right, and the parabola opens to the left; if $$a>0$$ then the vertex is on the left, and the parabola opens to the right.
• Vertex form of a vertical parabola: $$y=a{\left(x-h\right)}^2+k$$; if $$a<0$$ then the vertex is the highest point; if $$a>0$$ then then the vertex is the lowest point.

Points

• Vertex $$\left(h,k\right)$$: A parabola's vertex is the origin point of the parabola that is located between the directrix and the focus. The vertex form (see Vertex form) can be used to find the vertex of the parabola.
• Focus $$\left(h\pm p,k\right)\left(h,k\pm p\right)$$: A parabola's focus is a point located inside the curve of the parabola around which the parabola bends. The distances from the focus to any point on the parabola is equal to the distance from that point to the directrix.

Lines, line segments, and axes

• Axis of symmetry: A line that runs through the vertex of a parabola creating two congruent halves.
• Directrix: A line that runs perpendicular to the axis of symmetry of a parabola and parallel to the latus rectum. The distance from a vertex of a parabola to the directrix is the same as the distance from the vertex to the focus.
• Focal length $$\left(p\right)$$: The distance between a parabola's vertex and its focus is its focal length. This distance is equal to the distance from the vertex to the directrix.
• Latus rectum $$\left(4p\right)$$: A line segment inside a parabola that passes through the parabola's focus and is perpendicular to the axis of symmetry. The length of the latus rectum is equal to four times the focal length of the parabola, which is $$4p$$.