Characteristic of Interior Point of Circle whose Center is Origin
Jump to navigation
Jump to search
Theorem
Let $\CC$ be a circle of radius $r$ whose center is at the origin $O$ of a Cartesian plane.
Let $P = \tuple {x, y}$ be a point in the plane of $\CC$.
Then $P$ is in the interior of $\CC$ if and only if:
- $x^2 + y^2 - r^2 < 0$
Proof
Let $d$ be the distance of $P$ from $O$.
\(\ds d\) | \(=\) | \(\ds \sqrt {\paren {x - 0}^2 + \paren {y - 0}^2}\) | Distance Formula | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds d^2\) | \(=\) | \(\ds x^2 + y^2\) |
Then by definition of interior of $\CC$:
- $P$ is in the interior of $\CC$ if and only if $d^2 < r^2$
and the result follows.
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $10$.