Formation of Ordinary Differential Equation by Elimination/Examples/x^2 + y^2 equals a^2

Examples of Formation of Ordinary Differential Equation by Elimination
Consider the equation:


 * $(1): \quad x^2 + y^2 = a^2$

This can be expressed as the ordinary differential equation:


 * $\dfrac {\d y} {\d x} = -\dfrac x y$

which demonstrates that the radius of a circle where it meets the circle is perpendicular to the tangent at that point.

Proof
Differentiating $x$:

From Equation of Circle: Corollary $2$, $(1)$ is the equation of a circle whose center is at the origin.

The straight line $R$ from the origin to the point $\tuple {x, y}$ on the circle has slope $\dfrac y x$ by definition.

From Slope of Orthogonal Curves, a straight line $T$ through $\tuple {x, y}$ such that $\dfrac {\d y} {\d x} = -\dfrac x y$ is perpendicular to $R$.

But $T$ is by definition the tangent to the circle at $\tuple {x, y}$.

Hence we have shown that the radius of a circle where it meets the circle is perpendicular to the tangent at that point.