Shortest distance of an external point from a circle

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Shortest distance of an external point from a circle
Let us consider a point and a circle as shown in the below figure.


Let us join the point and centre of the circle and also let us draw a perpendicular line to the line segment joining the centre of the circle and the point as shown in the below figure.


We can find distance of the point from circle along different line segments joining the points on the circle and the considered point as shown in below figure.



We know that the shortest distance of a point from a line is the distance of the point from the line along the perpendicular line segment to the line.

Therefore in the figure, length of PQ line segment is the least among all other line segments lengths from the line.
Here all the line segments joining the considered point and other points on the circle other than PQ are obtained when the line segments from the considered point to the drawn line are extended as shown in the above figure.

In the above figure, length of PR is greater than the length of PQ.
As the length of PQ line segment is the least among all other line segments lengths from the line to the considered point, length of PQ is less than the lengths of line segments joining points on the circle and the considered point since the line segments joining points on the circle and the considered point are longer than the line segments from line to the considered point.

Therefore the length of line segment PQ will be than all the lengths of other line segments joining the considered point and points on the considered circle.
Therefore shortest distance of above considered point will be equal to the length of line segment PQ.
Therefore the shortest distance of a point from a circle will be along the line joining the considered point and the centre of the circle.


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