Equations of Normal to a Parabola – Explained with Examples | Testbook

Demystifying the Equations of Normal to a Parabola

In geometry and calculus, one important concept is the normal to a parabola. But what does that really mean?

Let’s break it down. A normal line to a parabola is simply a line that is at a right angle (90°) to the tangent line at a specific point on the parabola.

Now imagine a parabola given by the equation y² = 4ax. Pick any point P on this curve. If you draw a tangent at point P, that line just touches the curve without cutting it. The normal is the line that also passes through point P but goes perpendicular to the tangent.

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Here’s something important to remember:
The product of the slope of the tangent and the slope of the normal is always -1, since they are at right angles.

Types of Equations for Normals to a Parabola

You can write the equation of a normal line in three main ways:

1. Point Form – When you know the exact coordinates (x, y) on the parabola.
2. Parametric Form – When the point on the parabola is written using a parameter t, like (at², 2at).
3. Slope Form – When the slope of the normal line is known or assumed.

When you draw a normal line to a parabola, you're drawing a line that is perpendicular to the tangent at a particular point on the curve.

Let’s say you have a parabola and a point P(x₁, y₁) on it. The formula for the normal line at that point depends on the type and direction of the parabola.

Equation of Normal in Slope Form

Sometimes, instead of using a point on the parabola, we use the slope of the normal line (denoted by m) to write its equation. This is called the slope form of the normal to a parabola.

Let’s take the standard parabola y² = 4ax.

Parabola Equation	Point of Contact	Equation of Normal
y² = 4ax	(am², –2am)	y = mx – 2am – am³
y² = –4ax	(–am², 2am)	y = mx + 2am + am³
x² = 4ay	(–2a/m, a/m²)	y = mx + 2a + a/m²
x² = –4ay	(2a/m, –a/m²)	y = mx – 2a – a/m²

Equation of Normal in Parametric Form

When we use a parameter (usually t) to represent a point on a parabola, we can also write the normal to the parabola using this parameter. This form is called the parametric form of the normal.

Let’s start with the standard parabola:
y² = 4ax

Parabola Equation	Point on Parabola (Parametric Coordinates)	Equation of Normal
y² = 4ax	(at², 2at)	y = –tx + 2at + at³
y² = –4ax	(–at², 2at)	y = tx + 2at + at³
x² = 4ay	(2at, at²)	x = –ty + 2at + at³
x² = –4ay	(2at, –at²)	x = ty + 2at + at³

Solved Examples 

Let's look at some examples to better understand the concept of the equation of normal to a parabola.

Example 1

Find the equation of the normal to the parabola y² = 8x at the point (2, 4).

Solution:
The general form of the parabola is:
  y² = 4ax
Here, 4a = 8 ⇒ a = 2

For a point (x₁, y₁) on the parabola, the equation of the normal is:
  y - y₁ = - (y₁ / 2a)(x - x₁)

Substitute (x₁, y₁) = (2, 4) and a = 2:
  y - 4 = - (4 / 4)(x - 2)
  y - 4 = - (x - 2)
  y - 4 = -x + 2
  x + y = 6

 Final Answer: x + y = 6

Example 2

Find the normal to the parabola y² = 12x at the point where y = 6.

Solution:
Parabola is y² = 4ax → Here, 4a = 12 ⇒ a = 3
Let’s find the corresponding x for y = 6:

  y² = 12x ⇒ 36 = 12x ⇒ x = 3
Point on the parabola: (3, 6)

Use the normal formula:
  y - y₁ = - (y₁ / 2a)(x - x₁)
Substitute values:
  y - 6 = - (6 / 6)(x - 3)
  y - 6 = - (x - 3)
  y - 6 = -x + 3
  x + y = 9

 Final Answer: x + y = 9

Example 3

Find the normal to the parabola y² = 4x at the point (1, 2).

Solution:
This is already in standard form with 4a = 4 ⇒ a = 1

Use the normal equation:
  y - y₁ = - (y₁ / 2a)(x - x₁)

Substitute (1, 2):
  y - 2 = - (2 / 2)(x - 1)
  y - 2 = - (x - 1)
  y - 2 = -x + 1
  x + y = 3

 Final Answer: x + y = 3

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