The equation Ax^2 + 4y^2 = 16 represents an ellipse. Find all values of A such that its intersection with y = |x| has coordinates (x,y) which are integers.

To find all values of A such that the ellipse intersects with y = |x| at integer coordinates (x, y), we can substitute y = |x| into the equation of the ellipse and solve for A.

First, let's replace y in the equation Ax^2 + 4y^2 = 16 with |x|:

A*x^2 + 4(|x|)^2 = 16

Since y = |x|, we can rewrite the equation as:

A*x^2 + 4x^2 = 16

Combine like terms:

(A + 4)*x^2 = 16

Now, we want to find the values of A for which this equation has integer solutions for x. This means that the value inside the parentheses, (A + 4), must be a factor of 16.

The factors of 16 are: 1, 2, 4, 8, 16, -1, -2, -4, -8, -16

Therefore, the potential values for A are: -4, -2, -1, 0, 12, 14, 15

So, all values of A that satisfy the condition are -4, -2, -1, 0, 12, 14, and 15.