Passes though 3 5 and 5 3?

Answer 1

There are infinitely many possible answers.

For example, the hyperbola, xy = 15,

or a circle with centre (0,0) and radius = sqrt(34),

or more generally, any one of infinitely many circles - with centre (x,x) and radius = sqrt[(3-x)^2 + (5-x)^2], that is radius = sqrt(2x^2-16x+34).

There are straight lines, quadratic curves, cubic curves and so on, to polynomials whose degrees approach infinity.

Then there are trigonometric curves.