Problem 23

Given information

The given radius  line with slope intersects some of these circle.

To find: The smallest value of  such that any line with slope  intersect some of these circles.

Step-by-step explanation

Consider the line with slope   and passing through the origin.

Equation to the normal to the line   is  .

These two lines are shown in the following diagram. Green line represents  and brown line represents .

Figure  

From the figure it is observed that, the points with integer coordinates are  ……….etc.

Solve the equation  and  to solve point at which the line is tangent to the circle centered at.

Put into the equation .

 

 

 

 

Put into the equation .

 

 

 

 

Hence the point  is  .

Solve the equation and   to solve point  at which the line is tangent to the circle centered at  .

Put   in to the equation .

Put into the equation.

Hence the pointis

 

The slope of the line is given by the following equation.

Thus the slope of the line is, so.

Now solve this equation for.

 

Therefore, the minimum value ofisunits.