Properties of Normal to a Parabola

The set of points that are equidistant from the directrix (fixed line) and the focus (fixed point) is called a parabola. It possesses mirror symmetry and is U-shaped. y² = 4ax is the equation of the parabola. The axis of symmetry is defined as the line that is perpendicular to the fixed line called the directrix, and which passes through the fixed point called the focus. It cuts the parabola at the centre. Vertex is the point where the intersection of the parabola takes place through the axis of symmetry. It is also the point where the parabola is more sharply curved. The focal length is the distance between the vertex and the focus along the symmetrical axis. The latus rectum is called the chord of the parabola, as it is parallel to the fixed line called the directrix and passes through the focus.

Some important features of a parabola are as follows:

1] Vertex of a parabola is (0, 0).

2] The equation of the latus rectum is x = a.

3] The extremities of the latus rectum are given by (a, 2a), (a, – 2a).

4] The length of the latus rectum is 4a.

5] The focal distance is given by x + a.

6] The parametric form is x = at² and y = 2at, where t is the parameter.

7] The eccentricity of the parabola is equal to 1.

8] It is symmetric about the axis.

9] The tangents drawn to any particular point on the fixed line are perpendicular in nature.

10] The tangent present at the vertex is parallel to the fixed line.

Normal to the parabola: The topic of normal is strongly linked to the concept of tangents. The slope of tangent * slope of normal = – 1. The normal to a parabola can be represented in various ways. The equation of the normal at a point (x₁, y₁) is given by (y – y₁) = (- y₁ / 2a) (x – x₁).

Properties of normal to the parabola y² = 4ax are discussed below.

• Three normals can be constructed from a given point to the parabola y² = 4ax.
• The algebraic sum of the slope of the above normals is 0.
• The algebraic sum of the ordinates of the feet is 0.
• The centroid of the triangle is formed in such a way that their feet are joined, and it lies on the parabolic axis.
• The formula of the centroid thus formed is: ((am₁² + am₂² + am₃²) / 3, (2am₁ + 2am₂ + 2am₃) / 3) which is equal to (am₁² + am₂² + am₃²) / 3, 0).
• The tangent present at 1 extremity of a focal chord of a parabola lies parallel to the normal of another extremity.
• The normal that is other than the axis of the parabola doesn’t pass through the focus.
• The normals situated at the extremities of the latus rectum of the parabola intersect the parabolic axis at right angles.
• There exists one real normal, as the imaginary normals always are in pairs.
• The length of the normal is found using the formula a (t₁ – t₂) (t₁ + t₂)² + 4 = 4a (t₁² + 1)^3/2 / t₁².
• The point of intersection of the normals is (2a + a (t₁² + t₁t₂ + t₂²) – at₁t₂ (t₁ + t₂)).
• If the normals of the parabola y² = 4ax intersect again on the parabola at a different point, then t₁t₂ = 2 and t₃ = – (t₁ + t₂).

Conormal Points of a Parabola

The points on the curve at which the normal passes through the point that is common in nature are called co-normal points. They are also termed the feet of the normal. The sum of the ordinates of the points which are con-normal is 0. The centroid of the triangle, thus formed by the co-normal points, lies on the parabolic axis.

Properties of Normal to a Parabola – Video Lesson