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:

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.

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 $({x}_{0},{y}_{0})$ is given by: $\frac{{x}_{0}(x{x}_{0})}{{a}^{2}}\frac{{y}_{0}(y{y}_{0})}{{b}^{2}}=1$

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:

Definition:
 A normal line to a hyperbola at a given point is a straight line that is perpendicular to the tangent at that point.

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 $({x}_{0},{y}_{0})$ is given by: $\frac{y{y}_{0}}{x{x}_{0}}=\frac{{b}^{2}}{{a}^{2}}\cdot \frac{{x}_{0}}{{y}_{0}}$

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 $A{x}^{2}+B{y}^{2}+Cx+Dy+E=0$.

Tangent Equation:
 To find the tangent equation, differentiate the equation of the hyperbola implicitly and substitute the coordinates of the point $({x}_{0},{y}_{0})$ to get the slope.
 The tangent equation can then be expressed in the pointslope form: $y{y}_{0}=m(x{x}_{0})$.

Normal Equation:
 The normal equation is derived by finding the negative reciprocal of the slope of the tangent and expressing it in the pointslope form.
Parametric Equations:
For a hyperbola with parametric equations $x(t)$ and $y(t)$, the tangent and normal equations are: $\frac{x({t}_{0})(x{x}_{0})}{{a}^{2}}\frac{y({t}_{0})(y{y}_{0})}{{b}^{2}}=1$
$\frac{y{y}_{0}}{x{x}_{0}}=\frac{{b}^{2}}{{a}^{2}}\cdot \frac{x({t}_{0})}{y({t}_{0})}$
Translated Hyperbola (Center at $(h,k)$:

General Equation:
 For a hyperbola with center $(h,k)$: $\frac{(xh{)}^{2}}{{a}^{2}}\frac{(yk{)}^{2}}{{b}^{2}}=1$.

Tangent Equation:
 The tangent equation becomes: $\frac{({x}_{0}h)(x{x}_{0})}{{a}^{2}}\frac{({y}_{0}k)(y{y}_{0})}{{b}^{2}}=1$

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 $(3,2)$.
3. Tangent Equation (Standard Form):

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

Substituting ${x}_{0}=3$, ${y}_{0}=2$, ${a}^{2}=9$, and ${b}^{2}=4$, we get: $\frac{3(x3)}{9}\frac{2(y2)}{4}=1$
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: $\frac{y2}{x3}=\frac{4}{9}\cdot \frac{3}{2}$
5. Translated Hyperbola (Center at (h, k)):
 General form: $\frac{(xh{)}^{2}}{{a}^{2}}\frac{(yk{)}^{2}}{{b}^{2}}=1$
 For our hyperbola, let's consider $(h,k)=(0,0)$ for simplicity.
6. Tangent Equation (Translated Form):
 The tangent equation becomes: $\frac{{x}_{0}(x{x}_{0})}{{a}^{2}}\frac{{y}_{0}(y{y}_{0})}{{b}^{2}}=1$
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: $x(t)=3\mathrm{sec}(t)$ $y(t)=2\mathrm{tan}(t)$

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