Auxiliary Circle

The circle with the major axis as diameter. If Q(a cosθ, a sinθ) is on the auxiliary circle, the corresponding point P(a cosθ, b sinθ) on the ellipse has the same eccentric angle θ. The ellipse is a uniform vertical compression of the auxiliary circle by factor b/a.

The auxiliary circle of the ellipse $x^2/a^2 + y^2/b^2 = 1$ is:

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

Geometric connection: For any point $P(a\cos\theta, b\sin\theta)$ on the ellipse, the point $Q(a\cos\theta, a\sin\theta)$ lies on the auxiliary circle (same $x$-coordinate, $y$ scaled up by $a/b$).

The eccentric angle $\theta$ is the actual angle $\angle QOx$ for the corresponding point $Q$ on the auxiliary circle — not for $P$ on the ellipse.

The ellipse as a compression: The ellipse is obtained from the auxiliary circle by scaling all $y$-coordinates by $b/a$:

$$(x, y) \mapsto \left(x, \frac{b}{a}y\right)$$

This uniform compression preserves $x$-coordinates and multiplies all areas by $b/a$.

Area consequence: Area of ellipse $=$ Area of auxiliary circle $\times b/a = \pi a^2 \times (b/a) = \pi ab$.

The minor auxiliary circle: The circle $x^2 + y^2 = b^2$ (using the semi-minor axis) is called the minor auxiliary circle. It is less commonly used but appears in some locus problems.