\[x\]

and \[y\]

axes. We can find the distanceof the points of intersection of points on the \[x\]

axis and points on the \[y\]

axis by first finding the points of intersection, then using the distance formula for the distance between two points.Suppose we have the circle with equation given by

\[(x-2)^2+(y-4)^2=25\]

. This circle has centre \[(2,4)\]

and radius 5.The circle intersects the

\[x\]

axis when \[y=0\]

.\[(x-2)^2+(0-4)^2=25 \rightarrow x-2 =\pm \sqrt{25-16}= \pm 3 \rightarrow x=2-3=-1,2+3=5\]

.The points of intersection with the

\[x\]

axis are \[(-1,0),(5,0)\]

.The circle intersects the

\[y\]

axis when \[x=0\]

.\[(x-2)^2+(y-4)^2=25 \rightarrow y-4 =\pm \sqrt{25-4}= \pm \sqrt{21} \rightarrow y=4-\sqrt{21}, \: 4+\sqrt{21}\]

.The points of intersection with the

\[y\]

axis are \[(0, 4-\sqrt{21}),(0,4+\sqrt{21})\]

.The distance between the points closest to the origin,

\[(-1,0)\]

and \[(4-\sqrt{21},0)\]

is \[d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}=\sqrt{(-1-0)^2+(0-(4-\sqrt{21})^2}=1.157\]

to three decimal places.