Tangent to Astroid between Coordinate Axes has Constant Length

Theorem
Let $C_1$ be a circle of radius $b$ roll without slipping around the inside of a circle $C_2$ of radius $a = 4 b$.

Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin.

Let $P$ be a point on the circumference of $C_1$.

Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis.

Let $H$ be the astroid formed by the locus of $P$.

The segment of the tangent to $H$ between the $x$-axis and the $y$-axis is constant, immaterial of the point of tangency.

Proof

 * AstroidTangent.png

From Equation of Astroid, $H$ can be expressed as:


 * $\begin{cases}

x & = a \cos^3 \theta \\ y & = a \sin^3 \theta \end{cases}$

Thus the slope of the tangent to $H$ at $\tuple {x, y}$ is:

Thus the equation of the tangent to $H$ is given by:


 * $y - a \sin^3 \theta = -\tan \theta \paren {x - a \cos^3 \theta}$

The $x$-intercept is found by setting $y = 0$ and solving for $x$:

Similarly, the $y$-intercept is found by setting $x = 0$ and solving for $y$, which gives:
 * $y = a \sin \theta$

The length of the part of the tangent to $H$ between the $x$-axis and the $y$-axis is given by:

which is constant.