Question

What is the definition of tangent?

Answer 1

Hint: A tangent is defined for a particular curve. It is defined as the limiting case of a chord passing through the curve. When the two points on the curve, through which the chord passes, approach each other, the chord of the curve approaches the tangent to the curve.

Complete step by step solution:
Geometrically, a tangent is defined as that line which just touches a given curve at its point of tangency.
In the Calculus, a tangent is defined as the limiting chord passing through the two points, when they approach each other. Let us consider the below curve C, and choose two points A and B lying on it. We consider a chord passing through the curve at the points A and B.

Now, let the points A and B approach towards each other. For this, both of these points must travel down along the curve. After travelling some distance down, we get the chord AB as

Similarly, we make them approach towards each other, and finally they both will reach at the bottom point and the chord AB in this case will become the tangent to the curve C, as shown in the figure below.

Hence, we have defined the tangent for a curve.

Note: Most of the students understand the tangent by its geometrical definition, which states that it is a line which just touches a curve. But we must note that the geometrical definition is not sufficient. We must understand the tangent by its definition in the calculus, which states it as the limiting case of the chord.