Question 6

Let $$a$$ be an irrational number. How many lines through the point $$(a,2a)$$ contain at least two points with both coordinates rational?

Solution

A line containing two points with rational coordinates has rational slope. If such a line with slope $$m$$ passes through $$(a,2a)$$ and a rational point $$(h,k)$$, then $$k-2a=m(h-a)$$. Rearranging gives $$k-mh=(2-m)a$$, and because $$a$$ is irrational while the left side is rational, we must have $$m=2$$. Thus there is only one possible line.

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Let $$a$$ be an irrational number. How many lines through the point $$(a,2a)$$ contain at least two points with both coordinates rational?

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