Equation of a circle when the circle touches the both axes

Equation

$(1).\,\,$ $(x-a)^2+(y-a)^2$ $\,=\,$ $a^2$
$(2).\,\,$ $(x-b)^2+(y-b)^2$ $\,=\,$ $b^2$
$(3).\,\,$ $(x-r)^2+(y-r)^2$ $\,=\,$ $r^2$

Introduction

A circle that touches both horizontal and vertical axes of two dimensional Cartesian coordinate system can be expressed in mathematical form by an equation and it is called the equation of a circle when the circle is touching the both axes.

Let the coordinates of center (or centre) of a circle are denoted by $a$ and $b$, and the geometric coordinates of a point on the circumference of a circle are denoted by $x$ and $y$, and the radius of circle is denoted by $r$. Then, the equation of a circle, which touches the both horizontal $x$ axis and vertical $y$ axis of a two dimensional space is written as follows.

Simple form

$(1).\,\,$ $(x-a)^2+(y-a)^2$ $\,=\,$ $a^2$

$(2).\,\,$ $(x-b)^2+(y-b)^2$ $\,=\,$ $b^2$

$(3).\,\,$ $(x-r)^2+(y-r)^2$ $\,=\,$ $r^2$

Expansion

$(1).\,\,$ $x^2$ $+$ $y^2$ $-$ $2a(x+y)$ $+$ $a^2$ $\,=\,$ $0$

$(2).\,\,$ $x^2$ $+$ $y^2$ $-$ $2b(x+y)$ $+$ $b^2$ $\,=\,$ $0$

$(3).\,\,$ $x^2$ $+$ $y^2$ $-$ $2r(x+y)$ $+$ $r^2$ $\,=\,$ $0$

Other form

The equation of a circle, which touches the both horizontal axis and vertical axis is also written in the following form by taking $C(h, k)$ as the center or centre of a circle in coordinate form.

Simple form

$(1).\,\,$ $(x-h)^2+(y-h)^2$ $\,=\,$ $h^2$

$(2).\,\,$ $(x-k)^2+(y-k)^2$ $\,=\,$ $k^2$

$(3).\,\,$ $(x-r)^2+(y-r)^2$ $\,=\,$ $r^2$

Expansion

$(1).\,\,$ $x^2$ $+$ $y^2$ $-$ $2h(x+y)$ $+$ $h^2$ $\,=\,$ $0$

$(2).\,\,$ $x^2$ $+$ $y^2$ $-$ $2k(x+y)$ $+$ $k^2$ $\,=\,$ $0$

$(3).\,\,$ $x^2$ $+$ $y^2$ $-$ $2r(x+y)$ $+$ $r^2$ $\,=\,$ $0$

Proof

Learn how to derive the equation of a circle that touches the both horizontal and vertical axes.

Latest Math Problems

A best free mathematics education website for students, teachers and researchers.

Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Maths Problems

Learn how to solve the maths problems in different methods with understandable steps.

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.