Quadratic form upper bound

Question

Let $A$ be a symmetric matrix.
And we have the quadratic form:

$$(x-y)^TA(x-y)$$

Where we know that $||x - y||_2 \leq 1/3$ and $||x||_2, ||y||_2 \leq 1$

How can we derive an upper bound of the form:

$$|(x-y)^TA(x-y)| \leq c \cdot |x^TAx|$$