Tangent and Normal to an Ellipse

Tangent and Normal to an Ellipse

Tangent to an Ellipse:

1. Definition:

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

• The equation of the tangent to an ellipse at a point $\left({x}_{1},{y}_{1}\right)$ 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 semi-major axis, $b$ is the length of the semi-minor axis, and $\left({x}_{1},{y}_{1}\right)$ is the point of tangency.
• The equation of the tangent at a point $\left({x}_{1},{y}_{1}\right)$ on the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ is given by: $x{x}_{1}\frac{{a}^{2}}{{x}_{1}^{2}}+y{y}_{1}\frac{{b}^{2}}{{y}_{1}^{2}}={a}^{2}+{b}^{2}$
3. Properties:

• The slope of the tangent at a given point $\left({x}_{1},{y}_{1}\right)$ 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.
1. Slope:

• The slope of the tangent at a point $\left({x}_{1},{y}_{1}\right)$ on the ellipse is given by $\frac{-{b}^{2}{x}_{1}}{{a}^{2}{y}_{1}}$, where $a$ is the semi-major axis and $b$ is the semi-minor axis.
2. Intersection with Axes:

• The tangents drawn to an ellipse from a point outside the ellipse are the lines that intersect the x-axis and y-axis at the points where the ellipse intersects these axes.

Normal to an Ellipse:

1. Definition:

• A normal to an ellipse at a point $\left({x}_{1},{y}_{1}\right)$ on the curve is a line perpendicular to the tangent at that point.
2. Normal Equation:

• The equation of the normal to an ellipse at a point $\left({x}_{1},{y}_{1}\right)$ on the curve is given by: $\frac{x{x}_{1}}{{a}^{2}}+\frac{y{y}_{1}}{{b}^{2}}=\frac{{x}_{1}^{2}}{{a}^{2}}+\frac{{y}_{1}^{2}}{{b}^{2}}$
3. Properties of Normals:

1. Perpendicularity:

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

• The slope of the normal at a point $\left({x}_{1},{y}_{1}\right)$ on the ellipse is the negative reciprocal of the slope of the tangent at that point.
3. 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: $\frac{1}{4}+\frac{{y}^{2}}{9}=1$ $\frac{{y}^{2}}{9}=1-\frac{1}{4}=\frac{3}{4}$ ${y}^{2}=\frac{27}{4}$ $y=±\frac{3\sqrt{3}}{2}$

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 $y-\frac{3\sqrt{3}}{2}=-\frac{\sqrt{3}}{3}\left(x-1\right)$

And the equation of the normal line at the same point is $y-\frac{3\sqrt{3}}{2}=\frac{9\sqrt{3}}{4}\left(x-1\right)$