Auxiliary Circle

The concept of an auxiliary circle is primarily associated with the study of ellipses and hyperbolas in the field of conic sections. It is a useful tool for understanding certain properties of these curves and for solving related problems.

Definition

For an ellipse or hyperbola, the auxiliary circle is defined as the circle that has the same center as the ellipse or hyperbola and whose radius is equal to the length of the semi-major axis of the ellipse or hyperbola.

For an Ellipse

Given an ellipse with the equation:

$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$

where $a$ is the length of the semi-major axis and $b$ is the length of the semi-minor axis, the auxiliary circle is the circle with the equation:

$$ x^2 + y^2 = a^2 $$

For a Hyperbola

For a hyperbola with the equation:

$$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $$

the auxiliary circle is again the circle with the equation:

$$ x^2 + y^2 = a^2 $$

Importance

The auxiliary circle is particularly useful for:

• Parametric representation of the ellipse or hyperbola.
• Understanding the relationship between the conic section and trigonometric functions.
• Simplifying the process of finding tangents and normals to the ellipse or hyperbola.

Parametric Equations

Using the auxiliary circle, we can derive the parametric equations for an ellipse:

$$ \begin{align*} x &= a \cos \theta \ y &= b \sin \theta \end{align*} $$

where $\theta$ is the parameter, often interpreted as the angle formed by the radius of the auxiliary circle and the positive x-axis.

For a hyperbola, the parametric equations become:

$$ \begin{align*} x &= a \sec \theta \ y &= b \tan \theta \end{align*} $$

or, using hyperbolic functions:

$$ \begin{align*} x &= a \cosh t \ y &= b \sinh t \end{align*} $$

where $t$ is the hyperbolic angle.

Differences and Important Points

Here is a table summarizing the differences and important points regarding the auxiliary circle for an ellipse and a hyperbola:

Property	Ellipse	Hyperbola
Definition	Circle with radius equal to the semi-major axis of the ellipse.	Circle with radius equal to the semi-major axis of the hyperbola.
Equation	$x^2 + y^2 = a^2$	$x^2 + y^2 = a^2$
Use in Parametric Equations	$x = a \cos \theta$, $y = b \sin \theta$	$x = a \sec \theta$, $y = b \tan \theta$ or $x = a \cosh t$, $y = b \sinh t$
Relationship to Conic	Encloses the ellipse.	Intersects the hyperbola at its vertices.
Trigonometric Interpretation	$\theta$ is the angle in standard position.	$\theta$ or $t$ relates to the slope of the tangent line.

Examples

Example 1: Parametric Equations for an Ellipse

Given an ellipse with the equation $\frac{x^2}{9} + \frac{y^2}{4} = 1$, find the parametric equations using the auxiliary circle.

Solution:

The semi-major axis $a = 3$ and the semi-minor axis $b = 2$. The auxiliary circle has the equation $x^2 + y^2 = 9$. The parametric equations for the ellipse are:

$$ \begin{align*} x &= 3 \cos \theta \ y &= 2 \sin \theta \end{align*} $$

Example 2: Intersection of a Hyperbola and its Auxiliary Circle

Find the points of intersection between the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$ and its auxiliary circle.

Solution:

The semi-major axis $a = 4$. The auxiliary circle has the equation $x^2 + y^2 = 16$. To find the points of intersection, we solve the system of equations:

$$ \begin{align*} \frac{x^2}{16} - \frac{y^2}{9} &= 1 \ x^2 + y^2 &= 16 \end{align*} $$

Multiplying the first equation by 16 and adding it to the second equation, we get:

$$ \begin{align*} x^2 - \frac{16y^2}{9} + x^2 + y^2 &= 16 + 16 \ 2x^2 + \frac{y^2}{9} &= 32 \ y^2 &= 0 \end{align*} $$

Thus, $y = 0$ and $x = \pm 4$. The points of intersection are $(4, 0)$ and $(-4, 0)$.

In conclusion, the auxiliary circle is a powerful tool in the study of conic sections, providing a geometric and trigonometric framework for analyzing and solving problems related to ellipses and hyperbolas.