A lattice point is a point with integer coordinates. How many lattice points (x,y) with -100 less than or equal to x less than or equal to 100 and -100 less than or equal to y less than or equal to 100 are on the graph of the parametric equations x=30-40 cos t and y=-50 + 30 cos t?

Question

A lattice point is a point with integer coordinates. How many lattice points (x,y) with -100 less than or equal to x less than or equal to 100 and -100 less than or equal to y less than or equal to 100 are on the graph of the parametric equations x=30-40 cos t and y=-50 + 30 cos t?

Answer 1

the graph is an ellipse

(x-30)/40 = -cos t
(y+50)/30 = sin t

(x-30)^2/1600 + (y+50)^2/900 = 1

the ellipse has semi-axes 30 and 40

Not sure how many lattice points lie on the curve. At least 8.

Answer 2

To find the number of lattice points on the graph of the given parametric equations, we need to determine the values of t for which x and y are integers.

Let's start by re-arranging the first equation, x = 30 - 40cos(t):

cos(t) = (30 - x) / 40

Now, for x to be an integer, (30 - x) should be divisible by 40 (since cos(t) can only be in the range [-1,1]). Therefore, (30 - x) should have factors of 2 and 5 in its prime factorization:

(30 - x) = 2^a * 5^b where a >= 0 and b >= 0

Similarly, for y to be an integer, (-50 + y) should be divisible by 30:

(-50 + y) = 2^c * 3^d where c >= 0 and d >= 0

Now, let's substitute the second equation, y = -50 + 30cos(t), into the above expression for (-50 + y):

30cos(t) = 2^c * 3^d + 50

cos(t) = (2^c * 3^d + 50) / 30

To find the values of t for which cos(t) is rational, we need to find the values of c and d such that (2^c * 3^d + 50) is divisible by 30.

Now, both (2^c * 3^d + 50) and (30 - x) must be divisible by 2^a and 5^b. Hence, we need to find the common factors of (2^c * 3^d + 50) and (30 - x).

By analyzing the factors of (2^c * 3^d + 50) and (30 - x) for different values of a, b, c, and d, we can determine the possible values of a, b, c, and d that yield integer values for x and y.

This analysis can be done programmatically by iterating over the values of a, b, c, and d and checking if the conditions for x and y being integers are met. The total count of such lattice points will be the number of valid combinations of a, b, c, and d.

Note: As this involves a multi-layered calculation, it may be time-consuming to determine the exact number of lattice points algebraically. Hence, an alternative approach, such as computer programming or numerical estimation, would be more practical to obtain the result.