A parabola is a U-shaped plane curve where any point is at an equal distance from a fixed point, which is the focus, and from a fixed straight line, which is known as the directrix. The set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane is a parabola. In other words, a parabola is the shape formed when we throw a ball in the air. Let us learn the Parabola formula.
What is Parabola?
Parabola is a section of a right circular cone by a plane parallel to a generator of the cone. It is a locus of a point, which moves so that distance from a fixed point is equal to the distance from a fixed-line.
Equation of Parabola
y2 = x is the simplest equation of a parabola when the directrix is parallel to the y-axis.
If the directrix is parallel to the y-axis then the equation of a parabola is:
y2 = 4ax
|a||distance from the origin to the focus.|
If the parabola is sideways i.e., the directrix is parallel to the x-axis, the standard equation of a parabola becomes,
x2 = 4ay
Therefore, the equations of a parabola in all quadrants are given as:
- y2 = 4ax
- y2 = – 4ax
- x2 = 4ay
- x2 = – 4ay
Latus Rectum of Parabola: The chord that passes through the focus and is perpendicular to the axis of the parabola is the latus rectum of a parabola. Two parabolas are said to be equal if their latus rectum are equal.
Latus Ractum = 4a (length of latus Rectum)
Focal chord: Any chord passes through the focus of the parabola is a fixed chord or focal chord of the parabola.
The Focal Distance or directrix: The focal distance of any point p(x, y) on the parabola y2 = 4ax is the distance between point ‘p’ and focus.
Focal distance = x + a
Position of a point with respect to parabola
When, S1<0 (inside the curve)
S1=0 (on the curve)
S1>0 (outside the curve)
Q.1: The equation of a parabola is y2=16x. Find the equation of directrix, coordinates of the focus, and the length of the latus rectum.
Solution: The given equation is y2=16x.
Here, the coefficient of x is positive. Hence, the parabola opens towards the right.
On comparing this equation with y2=4ax,
we get, 4a=16a
Coordinates of the focus are (4, 0).
Equation of directix is x = −a,
x = −4.
Length of latus rectum = 4a = 4 x 4 = 16.
Q.2. Find the equation of the parabola whose coordinates of vertex and focus are (-2, 3) and (1, 3) respectively.
Solution: Given, co ordinates of vertex b= 3 , c = -2,
If the ordinates of vertex and focus are equal then the axis of the required parabola is parallel to x-axis.
a = abscissa of focus – abscissa of vertex
a = 1 – (- 2) = 1 + 2 = 3.
Therefore, the equation of the required parabola is
(y – b)2 = 4a (x – c)
(y – 3)2 = 4 × 3(x + 2)
y2 – 6y + 9 = 12x + 24
y2 – 6y – 12x = 15
The required equation of the parabola is y2 – 6y – 12x – 15 = 0