parabola jee mains notes boards

Parabola Equation

The general equation of a parabola is: y = a(x-h) ² + k or x = a(y-k) ² +h, where (h,k) denotes the vertex. The standard equation of a regular parabola is y ² = 4ax.

The general equation of a parabola is:

y = a(x - h) ² + k (regular)

x = a(y – k) ² + h (sideways)

Where,

(h, k) = vertex of the parabola

Focus: The focus of a parabola is a fixed point, typically denoted as (a, 0).

Directrix: The directrix is a fixed line, often represented as a vertical line passing through the point (-a, 0). The directrix is perpendicular to the axis of the parabola.

Focal Chord: A focal chord is a straight line that passes through the focus of the parabola. It intersects the parabola at two distinct points.

Focal Distance: The focal distance is the distance between a point on the parabola and the focus. It is also equal to the perpendicular distance from that point to the directrix.

Latus Rectum: The latus rectum is a focal chord that is perpendicular to the axis of the parabola. It passes through the focus and is characterized by its length, which is typically 4a. The endpoints of the latus rectum are (a, 2a) and (a, -2a).

Eccentricity: The eccentricity (e) of a parabola is always equal to 1. It represents the ratio of the distance of a point on the parabola from the focus to the distance of that point from the directrix.

These terms and features are fundamental for understanding and working with parabolas in various mathematical and practical applications, providing insights into their characteristics and properties.