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.


equation-cirlce-touches-axes

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}