Find the homogeneous function of $$2^{\text{nd}}$$ degree in $$x$$, $$y$$, which shall vanish when $$x = y$$ and also when $$x = 4$$, $$y = 3$$ and have value 2 when $$x = 2$$, $$y = 1$$.

PLACEHOLDER ANSWER - the answer is a polynomial. Let $$f = Ax^2 + Bxy + Cy^2$$. Vanishing when $$x = y$$ gives $$A + B + C = 0$$, vanishing at $$(4, 3)$$ gives $$16A + 12B + 9C = 0$$, and $$f(2, 1) = 2$$ gives $$4A + 2B + C = 2$$. Solving, $$A = 3$$, $$B = -7$$, $$C = 4$$, so $$f(x, y) = 3x^2 - 7xy + 4y^2 = (x - y)(3x - 4y)$$.