# 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 $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ at the point $\left({x}_{0},{y}_{0}\right)$ is given by: $\frac{{x}_{0}\left(x-{x}_{0}\right)}{{a}^{2}}-\frac{{y}_{0}\left(y-{y}_{0}\right)}{{b}^{2}}=1$
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 $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ at the point $\left({x}_{0},{y}_{0}\right)$ is given by: $\frac{y-{y}_{0}}{x-{x}_{0}}=-\frac{{b}^{2}}{{a}^{2}}\cdot \frac{{x}_{0}}{{y}_{0}}$
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 $A{x}^{2}+B{y}^{2}+Cx+Dy+E=0$.
1. Tangent Equation:

• To find the tangent equation, differentiate the equation of the hyperbola implicitly and substitute the coordinates of the point $\left({x}_{0},{y}_{0}\right)$ to get the slope.
• The tangent equation can then be expressed in the point-slope form: $y-{y}_{0}=m\left(x-{x}_{0}\right)$.
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 $x\left(t\right)$ and $y\left(t\right)$, the tangent and normal equations are: $\frac{x\left({t}_{0}\right)\left(x-{x}_{0}\right)}{{a}^{2}}-\frac{y\left({t}_{0}\right)\left(y-{y}_{0}\right)}{{b}^{2}}=1$

$\frac{y-{y}_{0}}{x-{x}_{0}}=-\frac{{b}^{2}}{{a}^{2}}\cdot \frac{x\left({t}_{0}\right)}{y\left({t}_{0}\right)}$

Translated Hyperbola (Center at $\left(h,k\right)$:

1. General Equation:

• For a hyperbola with center $\left(h,k\right)$: $\frac{\left(x-h{\right)}^{2}}{{a}^{2}}-\frac{\left(y-k{\right)}^{2}}{{b}^{2}}=1$.
2. Tangent Equation:

• The tangent equation becomes: $\frac{\left({x}_{0}-h\right)\left(x-{x}_{0}\right)}{{a}^{2}}-\frac{\left({y}_{0}-k\right)\left(y-{y}_{0}\right)}{{b}^{2}}=1$
3. Normal Equation:

• The normal equation becomes: $\frac{y-{y}_{0}}{x-{x}_{0}}=-\frac{{b}^{2}}{{a}^{2}}\cdot \frac{{x}_{0}-h}{{y}_{0}-k}$

Example:

Consider the hyperbola given by the equation:

$\frac{{x}^{2}}{9}-\frac{{y}^{2}}{4}=1$

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

1. Standard Form Hyperbola:

• Standard form: $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$
• For our hyperbola, ${a}^{2}=9$ and ${b}^{2}=4$.

2. Choose a Point on the Hyperbola:

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

3. Tangent Equation (Standard Form):

• The equation of the tangent line is: $\frac{{x}_{0}\left(x-{x}_{0}\right)}{{a}^{2}}-\frac{{y}_{0}\left(y-{y}_{0}\right)}{{b}^{2}}=1$

• Substituting ${x}_{0}=3$, ${y}_{0}=2$, ${a}^{2}=9$, and ${b}^{2}=4$, we get: $\frac{3\left(x-3\right)}{9}-\frac{2\left(y-2\right)}{4}=1$

4. Normal Equation (Standard Form):

• The equation of the normal line is: $\frac{y-{y}_{0}}{x-{x}_{0}}=-\frac{{b}^{2}}{{a}^{2}}\cdot \frac{{x}_{0}}{{y}_{0}}$

• Substituting the values, we get: $\frac{y-2}{x-3}=-\frac{4}{9}\cdot \frac{3}{2}$

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

• General form: $\frac{\left(x-h{\right)}^{2}}{{a}^{2}}-\frac{\left(y-k{\right)}^{2}}{{b}^{2}}=1$
• For our hyperbola, let's consider $\left(h,k\right)=\left(0,0\right)$ for simplicity.

6. Tangent Equation (Translated Form):

• The tangent equation becomes: $\frac{{x}_{0}\left(x-{x}_{0}\right)}{{a}^{2}}-\frac{{y}_{0}\left(y-{y}_{0}\right)}{{b}^{2}}=1$

7. Normal Equation (Translated Form):

• The normal equation becomes: $\frac{y-{y}_{0}}{x-{x}_{0}}=-\frac{{b}^{2}}{{a}^{2}}\cdot \frac{{x}_{0}-h}{{y}_{0}-k}$

8. Parametric Form:

• Let's express the hyperbola parametrically: $x\left(t\right)=3\mathrm{sec}\left(t\right)$  $y\left(t\right)=2\mathrm{tan}\left(t\right)$

• For tangent and normal, use: $\frac{x\left({t}_{0}\right)\left(x-{x}_{0}\right)}{{a}^{2}}-\frac{y\left({t}_{0}\right)\left(y-{y}_{0}\right)}{{b}^{2}}=1$

$\frac{y-{y}_{0}}{x-{x}_{0}}=-\frac{{b}^{2}}{{a}^{2}}\cdot \frac{x\left({t}_{0}\right)}{y\left({t}_{0}\right)}$