Question

The length of a tangent, drawn from a point A to a circle, is `4/3` of the radius r. What is the shortest distance from A to the circle ?

Answer 1

`color(blue)(2/3r)`

Explanation:

From the diagram.

`AB = 4/3r`

The angle bisector of `BAC` always passes through the centre of the circle. This is the line `AO`. Since the radius always forms an angle of `90^@` at the point of tangency. `/_ABO` and `/_ACO=90^@`

By Pythagoras' theorem:

`AO=sqrt((4/3r)^2+r^2`

The shortest distance is the line `AD`

Since `DO=r`

`AD=AO-DO=sqrt((4/3r)^2+r^2)-r`

`AD=sqrt(r^2(25/9))-r=5/3sqrt(r^2)-r`

For `|sqrt(r^2)|`

`=5/3r-r=2/3r`