# Equation of a Circle Touching Both Axes

In this tutorial we find the equation of circles with both axes touching, i.e. the X-axis and Y-axis, with any given radius. So we will find the equation of a circle in all four quadrants.

Let the equation of the required circle having a center and radius be

\[{\left( {x – h} \right)^2} + {\left( {y – k} \right)^2} = {r^2}\,\,\,{\text{ – – – }}\left( {\text{i}} \right)\]

**In the First Quadrant:**

In the first quadrant, the equation of a circle can be found by using center $${C_1}\left( {r,r} \right)$$ and the radius is equal to $$r$$, so equation (i) becomes

\[{\left( {x – r} \right)^2} + {\left( {y – r} \right)^2} = {r^2}\]

**In the Second Quadrant:**

In the second quadrant, the equation of a circle can be found by using center $${C_2}\left( { – r,r} \right)$$ and the radius is equal to $$r$$, so equation (i) becomes

\[{\left( {x + r} \right)^2} + {\left( {y – r} \right)^2} = {r^2}\]

**In the Third Quadrant:**

In the third quadrant, the equation of a circle can be found by using center $${C_3}\left( { – r, – r} \right)$$ and the radius is equal to $$r$$, so equation (i) becomes

\[{\left( {x + r} \right)^2} + {\left( {y + r} \right)^2} = {r^2}\]

**In the Forth Quadrant:**

In the forth quadrant, the equation of a circle can be found by using center $${C_4}\left( {r, – r} \right)$$ and the radius is equal to $$r$$, so equation (i) becomes

\[{\left( {x – r} \right)^2} + {\left( {y + r} \right)^2} = {r^2}\]