Derivation of Parabola Equation
Take a point P with coordinates (x, y) on the parabola which lies on the X-Y plane. By the definition of the parabola, the distance of any point on the parabola from the focus and from the directrix is equal.
Now distance of P from the directrix is given by PB where the coordinates of B are (-a, y) as it lies on the directrix, and the distance of P from focus is PF.
Image of parabola is shown below,
By the definition of parabola, PF = PB . . . .(1)
Using Distance Formula, we get
PF = √(x-a)2+(y-0)2= √{(x-a)2+y2} . . . .(2)
PB = √{(x+a)2} . . . .(3)
By using, equations (1), (2), and (3), we get
√{(x-a)2+y2} = √{(x+a)2}
⇒ (x – a)2 + y2 = (x + a)2
⇒ x2 + a2 – 2ax + y2 = x2 + a2 + 2ax
⇒ y2 – 2ax = 2ax
y2 = 4ax
Which is the required equation of the parabola.
Similarly, the equation for other parabolas i.e., x2 = 4ay, y2 = -4ax, and x2 = -4ay, can also be proved.
Parabola – Graph, Properties, Examples & Equation of Parabola
Parabola is one of the conic sections in Math. It is an intersection of a surface plane and a double-napped cone. A parabola is a U-shaped curve that can be either concave up or down, depending on the equation. Parabolic curves are widely used in many fields such as physics, engineering, finance, and computer sciences.
In this article, we will understand what is a Parabola, its graph, Parabola properties, Parabola examples, and equation of parabola in detail below.
Table of Content
- What is Parabola in Maths?
- Parabola Definition
- Parabola Shape
- Parabola Equation
- Properties of Parabola
- Standard Equation of Parabola
- Important Terms Related to Parabola
- Derivation of Parabola Equation
- Graph of Parabola
- Position of Point Relative to the Parabola
- Intersection with Straight Line
- General Equations of Parabola
- Parametric Coordinates of a Parabola
- Equation of Tangent to a Parabola
- Equation of Tangent in Point Form
- Equation of Tangent in Parametric Form
- Equation of Tangent in Slope Form
- Pair of Tangent from an External Point
- Director Circle of Parabola
- Chord of Contact
- Equation of Normal to a Parabola
- Equation of Normal in Slope Form
- Equation of Normal in Point Form
- Equation of Normal in Parametric Form
- Parabola Formulas
- Parabola Solved Examples
- Practice Questions on Parabola