Tangent and Normal of a Hyperbola

I. Introduction to Tangent and Normal:

  • In differential calculus, the concepts of tangent and normal are crucial for understanding the local behavior of curves. For a hyperbola, these concepts play a significant role.

II. Tangent Line to a Hyperbola:

  1. Definition:

    • A tangent line to a hyperbola at a given point is a straight line that touches the hyperbola at that point and has the same slope as the hyperbola at that specific point.
  2. Tangent Equation:

    • The equation of the tangent line to the hyperbola x2a2y2b2=1 at the point (x0,y0) is given by: x0(xx0)a2y0(yy0)b2=1
  3. Slope of Tangent:

    • The slope of the tangent line to the hyperbola at a given point is equal to the derivative of the hyperbola's equation at that point.

III. Normal Line to a Hyperbola:

  1. Definition:

    • A normal line to a hyperbola at a given point is a straight line that is perpendicular to the tangent at that point.
  2. Normal Equation:

    • The equation of the normal line to the hyperbola x2a2y2b2=1 at the point (x0,y0) is given by: yy0xx0=b2a2x0y0
  3. Slope of Normal:

    • The slope of the normal line to the hyperbola at a given point is the negative reciprocal of the slope of the tangent at that point.

IV. Geometric Interpretation:

  • The tangent and normal lines provide insights into the instantaneous rate of change and perpendicular direction at a specific point on the hyperbola.

General Form of Hyperbola:

  • A hyperbola in general form is given by Ax2+By2+Cx+Dy+E=0.
  1. Tangent Equation:

    • To find the tangent equation, differentiate the equation of the hyperbola implicitly and substitute the coordinates of the point (x0,y0) to get the slope.
    • The tangent equation can then be expressed in the point-slope form: yy0=m(xx0).
  2. Normal Equation:

    • The normal equation is derived by finding the negative reciprocal of the slope of the tangent and expressing it in the point-slope form.

Parametric Equations:

For a hyperbola with parametric equations x(t) and y(t), the tangent and normal equations are: x(t0)(xx0)a2y(t0)(yy0)b2=1

 yy0xx0=b2a2x(t0)y(t0)

Translated Hyperbola (Center at (h,k):

  1. General Equation:

    • For a hyperbola with center (h,k): (xh)2a2(yk)2b2=1.
  2. Tangent Equation:

    • The tangent equation becomes: (x0h)(xx0)a2(y0k)(yy0)b2=1
  3. Normal Equation:

    • The normal equation becomes: yy0xx0=b2a2x0hy0k

Example: 

Consider the hyperbola given by the equation:

x29y24=1

Let's find the equations of the tangent and normal lines at a specific point on this hyperbola.

1. Standard Form Hyperbola:

  • Standard form: x2a2y2b2=1
  • For our hyperbola, a2=9 and b2=4.

2. Choose a Point on the Hyperbola:

  • Let's choose a point on the hyperbola, say (3,2).

3. Tangent Equation (Standard Form):

  • The equation of the tangent line is: x0(xx0)a2y0(yy0)b2=1

  • Substituting x0=3, y0=2, a2=9, and b2=4, we get: 3(x3)92(y2)4=1

4. Normal Equation (Standard Form):

  • The equation of the normal line is: yy0xx0=b2a2x0y0

  • Substituting the values, we get: y2x3=4932

5. Translated Hyperbola (Center at (h, k)):

  • General form: (xh)2a2(yk)2b2=1
  • For our hyperbola, let's consider (h,k)=(0,0) for simplicity.

6. Tangent Equation (Translated Form):

  • The tangent equation becomes: x0(xx0)a2y0(yy0)b2=1

7. Normal Equation (Translated Form):

  • The normal equation becomes: yy0xx0=b2a2x0hy0k

8. Parametric Form:

  • Let's express the hyperbola parametrically: x(t)=3sec(t)  y(t)=2tan(t)

  • For tangent and normal, use: x(t0)(xx0)a2y(t0)(yy0)b2=1

 yy0xx0=b2a2x(t0)y(t0)