Equation of Parabola Derivation

Let P be a point on the parabola whose coordinates are (x, y). From the definition of a parabola, the distance of point P to the focus (F) is equal to the distance of the same point P to the directrix of a parabola. Now, let us consider a point X on the directrix, whose coordinates are (-a, y).

From the definition of the eccentricity of a parabola, we have 

e = PF/PX = 1

⇒ PF = PX

The coordinates of the focus are (a, 0). Now, by using the coordinate distance formula, we can find the distance of point P (x, y) to the focus F (a, 0).

PF = √[(x – a)² + (y – 0)²]

⇒ PF = √[(x – a)² + y²]   —————— (1)

The equation of the directrix is x + a = 0. To find the distance of PX, we use the perpendicular distance formula.

PX = (x + a)/√[1² + 0²]

⇒ PX = x +a    —————— (2)

We already know that PF = PX. So, equate equations (1) and (2).

√[(x – a)² + y²] = (x + a)

By, squaring on the both sides we get,

⇒ [(x – a)² + y²] = (x + a)²

⇒ x² + a² – 2ax + y² = x² + a² + 2ax

⇒ y² – 2ax = 2ax

⇒ y² = 2ax + 2ax ⇒ y² = 4ax

Thus, we have derived the equation of a parabola. Similarly, we can derive the standard equations of the other three parabolas.

• y² = -4ax
• x² = 4ay
• x² = -4ay

y² = 4ax, y² = -4ax, x² = 4ay, and x² = -4ay are the standard equations of a parabola.

Articles Related to Parabola:

• Equation of Circle
• Equation of Ellipse
• Equation of Hyperbola
• Applications of Parabola in Real Life

Standard form of a parabola is y = ax² + bx + c where a, b, and c are real numbers and a is not equal to zero. A parabola is defined as the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane.

In this article, we will understand what is a Parabola, the standard equation of a Parabola, related examples and others in detail.