The subtangent at a point on a curve is the projection of the tangent line onto the x-axis. It represents the horizontal distance between the point of tangency and the point where the tangent intersects the x-axis.
The subnormal at a point on a curve is the projection of the normal line onto the x-axis. It represents the horizontal distance between the point of tangency and the point where the normal intersects the x-axis.
For a curve y = f(x) at a point P(x₀, y₀), the lengths are calculated as follows:
| Quantity | Formula |
| Tangent | Length of Tangent = y₀ √(1 + (dx/dy)²) |
| Subtangent | Length of Subtangent = |y₀ × (dx/dy)| |
| Normal | Length of Normal = y₀ √(1 + (dy/dx)²) |
| Subnormal | Length of Subnormal = |y₀ × (dy/dx)| |
Consider the standard parabola y² = 4ax. At a point P(x₁, y₁) on the parabola, the lengths are:
| Quantity | Formula |
| Subtangent | Length of Subtangent = 2x₁ |
| Subnormal | Length of Subnormal = a |
Problem: Find the lengths of the tangent, subtangent, normal, and subnormal to the curve y² = 4ax at the point (at², 2at).
Solution:
Equation of the Tangent:
The equation of the tangent to the parabola y² = 4ax at (at², 2at) is:
y = tx + at
Length of the Tangent:
Using the formula:
Length of Tangent = 2at √(1 + t²)
Length of the Subtangent:
Length of Subtangent = 2a
Length of the Subnormal:
Length of Subnormal = 2at²
Additional Example
Problem: For the curve y = e^x, find the length of the subtangent at any point (x₀, y₀).
Solution:
Derivative:
dy/dx = e^x
Length of Subtangent:
Length of Subtangent = 1
The subtangent is the horizontal projection of the tangent on the x-axis, while the subnormal is the horizontal projection of the normal on the x-axis. These quantities help in analyzing the geometric properties of curves.
Subtangent and subnormal are often used to calculate distances and slopes related to tangents and normals of curves, which are critical for solving advanced calculus problems in JEE.
These concepts are commonly applied in deriving equations of tangents and normals, optimizing distances, and solving problems involving curve analysis, especially for parabolas, ellipses, and hyperbolas.
Subtangent and subnormal lengths are typically expressed as absolute values, as they represent distances. However, the sign may indicate direction when dealing with coordinate geometry.
Parabolas have specific formulas for subtangent and subnormal that simplify their calculation, making these concepts essential for understanding the geometric behavior of parabolic curves in competitive exams like JEE.
