Tangent and Normal to an Ellipse
Tangent and Normal to an Ellipse
Tangent to an Ellipse:

Definition:
 A tangent to an ellipse is a straight line that intersects the ellipse at only one point, touching the curve without crossing it.

Tangent Equation:
 The equation of the tangent to an ellipse at a point $({x}_{1},{y}_{1})$ on the curve is given by: $\frac{x{x}_{1}}{{a}^{2}}+\frac{y{y}_{1}}{{b}^{2}}=1$ Where $a$ is the length of the semimajor axis, $b$ is the length of the semiminor axis, and $({x}_{1},{y}_{1})$ is the point of tangency.
 The equation of the tangent at a point $({x}_{1},{y}_{1})$ on the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ is given by: $x{x}_{1}\frac{{a}^{2}}{{x}_{1}^{2}}+y{y}_{1}\frac{{b}^{2}}{{y}_{1}^{2}}={a}^{2}+{b}^{2}$

Properties:
 The slope of the tangent at a given point $({x}_{1},{y}_{1})$ on the ellipse can be found using the derivative of the equation of the ellipse.
 The slope of the tangent is perpendicular to the radius vector of the ellipse at the point of tangency.

Point of Tangency:
 A tangent to an ellipse touches the ellipse at a single point.

Slope:
 The slope of the tangent at a point $({x}_{1},{y}_{1})$ on the ellipse is given by $\frac{{b}^{2}{x}_{1}}{{a}^{2}{y}_{1}}$, where $a$ is the semimajor axis and $b$ is the semiminor axis.

Intersection with Axes:
 The tangents drawn to an ellipse from a point outside the ellipse are the lines that intersect the xaxis and yaxis at the points where the ellipse intersects these axes.
Normal to an Ellipse:

Definition:
 A normal to an ellipse at a point $({x}_{1},{y}_{1})$ on the curve is a line perpendicular to the tangent at that point.

Normal Equation:
 The equation of the normal to an ellipse at a point $({x}_{1},{y}_{1})$ on the curve is given by: $\frac{x{x}_{1}}{{a}^{2}}+\frac{y{y}_{1}}{{b}^{2}}=\frac{{x}_{1}^{2}}{{a}^{2}}+\frac{{y}_{1}^{2}}{{b}^{2}}$

Properties of Normals:

Perpendicularity:
 The normal to an ellipse at a point is perpendicular to the tangent at that same point.

Slope:
 The slope of the normal at a point $({x}_{1},{y}_{1})$ on the ellipse is the negative reciprocal of the slope of the tangent at that point.

Intersection with Axes:
 The normals drawn to an ellipse from a point outside the ellipse intersect the axes at the points where the ellipse intersects these axes.

Example :
Let's take an ellipse with the equation: $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1$
This ellipse has a horizontal major axis (2a = 6, so a = 3 and 2b = 4, so b = 2).
At any point (x, y) on the ellipse, the slope of the tangent is given by $\frac{{b}^{2}x}{{a}^{2}y}$, and the slope of the normal is $\frac{ay}{bx}$.
Let's find the tangent and normal lines at a specific point on the ellipse, for example, when x = 1.
When x = 1, we can find y by substituting it into the equation of the ellipse: $\frac{1}{4}+\frac{{y}^{2}}{9}=1$ $\frac{{y}^{2}}{9}=1\frac{1}{4}=\frac{3}{4}$ ${y}^{2}=\frac{27}{4}$ $y=\pm \frac{3\sqrt{3}}{2}$
So, the point on the ellipse is (1, $\frac{3\sqrt{3}}{2}$).
Now, let's calculate the slope of the tangent and normal lines at this point:
Tangent slope: ${m}_{t}=\frac{{b}^{2}x}{{a}^{2}y}=\frac{4\cdot 1}{9\cdot \frac{3\sqrt{3}}{2}}=\frac{4}{6\sqrt{3}}=\frac{\sqrt{3}}{3}$
Normal slope: ${m}_{n}=\frac{ay}{bx}=\frac{3\cdot \frac{3\sqrt{3}}{2}}{2\cdot 1}=\frac{9\sqrt{3}}{4}$
Therefore, the equation of the tangent line at the point (1, $\frac{3\sqrt{3}}{2}$) is $y\frac{3\sqrt{3}}{2}=\frac{\sqrt{3}}{3}(x1)$
And the equation of the normal line at the same point is $y\frac{3\sqrt{3}}{2}=\frac{9\sqrt{3}}{4}(x1)$