We say that a point p = (x, y) in r2 is rational if both x and y are rational. more precisely, p is rational if p = (x, y) ∈ q2. an equation f(x, y) = 0 is said to have a rational point if there exists x0, y0 ∈ q such that f(x0, y0) = 0. for example, the curve x2+y2−1=0 has rational point (x0,y0)=(1,0). show that the curve x2 + y2 − 3 = 0 has no rational points

The first curve is the unit circle, which has many rational points, each corresponding to a Pythagorean Triple. To check (1,0) we substitute

That's a rational point, albeit a boring one. The first Pythagorean Triple on the list gives a more interesting point, (x,y)=(3/5, 4/5), which is also on the curve.

Let's assume there are rational points; we write them with a common denominator. Let x=a/d, y=b/d

There are three cases:

Case 1: 3 divides exactly one of a or b

Case 2: 3 divides both a and b

Case 3: 3 divides neither a nor b

In case 1, let's say without loss of generality that it's a that's a multiple of 3. So is of course. But if b is not a multiple of 3, i.e for some natural n, its square isn't a multiple of three either:

We see the square of a non-multiple of 3 necessarily has a remainder of 1 when divided by three.

So the sum of squares isn't a multiple of 3 so can't be . That shows case 1 can't be the case.

In case 2, 3 divides both a and b, so each of the squares will be a power of 9 times some other factors, i.e. each square has an even number of factors of 3. So will the sum, because the other non-multiple of three factors when squared and added won't be a multiple of 3 as we showed in case 1. So d will have half of those factors of 3 and since there were an even number there won't be any leftover for the outside factor of 3. So case 2 can't be true.

Case 3. We showed both squares have remainder 1 when divided by 3, so their sum will have remainder 2, so can't be a multiple of 3.

None of the possibilities can give rise to rational points so we conclude there are no rational points on the curve.

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is symmetric about the line y = 120,000.

step-by-step explanation: