Properties of the Focal Chord, Tangent, and Normal of Parabola

There are several interesting properties of the focal chords and the properties of a parabola. Let’s start with the length. The length is the distance from the center of the circle to the chord. The focal length is the length of the focal chord. This is one of the important properties of the parabola.because it determines how sharply or broadly the image will be focused.

The other important property is the angle. This is the angle between the tangent and the chord at the point of contact. This angle determines how much light will be reflected off of the surface.

Parabola

Before diving into the properties of the tangent and normal of parabola, let’s take a look at the properties of a parabola.

• A parabola is a conic section that has the property that the distance from any point on the curve to a fixed point, called the focus, is always the same. 
• The line perpendicular to the directrix and passing through the focus is called the axis of symmetry. This is one of the important properties of a parabola.
• The point where the axis of symmetry intersects the parabola is called the vertex. A parabola can open up, down, left, or right depending on the sign of the parameter c. If c is positive, the parabola will open to the right. If c is negative, the parabola will open to the left. Such are the properties of a parabola.

Focal Chord of the Parabola and the Important Properties of the Focal Chord

After the properties of a parabola, let’s study the focal chord.

• The chord which passes through the focus is called the focal chord of the parabola.
• The focal distance of some point P which is on the parabola with equation y2  = 4ax will be the distance between the point P and the focus.

Tangent of Parabola

The information about the tangent of parabola is as given below:

• The tangent to a conic section is a straight line that intersects the curve at one point. 
• The tangent is perpendicular to the chord at the point of intersection.
•  The length of the tangent is the distance from the point of intersection to the focus. The angle between the tangent and chord is called the angle of incidence. This angle is important because it determines how much light will be reflected off of the surface.

Important Properties of Tangent

Now let’s move on to the important properties of tangent. 

• The first property is the angle. This is the angle between the tangent and the chord at the point of contact. This angle is important because it determines how much light will be reflected off of the surface.
• The second property is the length. This is the distance from the center of the circle to the point of contact. This is important because it determines how sharply or broadly the image will be focused.
• The last property is the curvature. This is how curved the tangent is at the point of contact. This is important because it determines how much light will be reflected off of the surface.

The Normal to a Parabola

The information about the normal to parabola is given below.

• The normal to a conic section is a straight line that intersects the curve at one point. The normal is perpendicular to the tangent at the point of intersection. 
• The length of the normal is the distance from the point of intersection to the center of the curve. 
• The angle between the normal and chord is called the angle of refraction. This angle is important because it determines how much light will be reflected off of the surface.

Important Properties of Normal of a Parabola

Finally, let’s talk about the properties of the normal.

•  The first property is the angle. This is the angle between the normal and the chord at the point of contact. This angle is important because it determines how much light will be reflected off of the surface.
• The second property is the length. This is the distance from the center of the circle to the point of contact. This is important because it determines how sharply or broadly the image will be focused.
• The last property is the curvature. This is how curved the normal is at the point of contact. This is important because it determines how much light will be reflected off of the surface. 

Conclusion

The curvature of a conic section is the rate of change of the tangent vector with respect to arc length. It is a measure of how sharply or broadly the image is focused. The curvature is zero at the focus and increases as we move away from the focus.  The focal chord of the parabola passes through the focus of the parabola.

The line that touches the parabola at exactly one line is called the tangent of the parabola. The line that is perpendicular to the tangent at the point of contact is the line normal to the parabola.