   Chapter 2.P, Problem 25P

Chapter
Section
Textbook Problem

# A lattice point in the plane is a point with integer coordinates. Suppose that circles with radius r are drawn using all lattice points as centers. Find the smallest value of r such that any line with slope 2 5 intersects some of these circles.

To determine

To find:

The smallest value of r such that any line with slope 25 intersects some of these circles

Solution:The smallest value of r such that any line with slope 25 intersects some of these circles is, r0.093

Explanation

1) Formula:

i. Tangent normal formula: If m1 is the slope of tangent and m2 is the slope of normal then,

m1*m2=-1

ii. Equation of line passing through the origin is,

y=mx

iii. Point-slope formula:

m=y-y1x-x1

2) Given:

Slope of Tangent line is 25

3) Calculation:

By using tangent normal formula, the slope of the normal line is,

m1*m2=-1

25m2=-1

m2=-52

Equation of line passing through the origin is, y=mx

Therefore,

y=-52x

Now, equation of circle centered at origin is, x2+y2=r2

Solving these equations,

x2+-52x2=r2

x2+254x2=r2

294x2=r2

x2=429r2

x=229r

Substitute in y=-52x

Therefore,

y=-52229r=-529r

Because of periodic nature of lattice points, it suffices to consider points in 5/2 grid shown

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