1. The problem is to solve or analyze the equation $ (x - y)^2 = 2ax $. 2. This is an algebraic equation involving $x$ and $y$ with a parameter $a$. We can expand and simplify it to understand its form better. 3. Expand the left side using the formula $(A - B)^2 = A^2 - 2AB + B^2$: $$ (x - y)^2 = x^2 - 2xy + y^2 $$ 4. Substitute back into the equation: $$ x^2 - 2xy + y^2 = 2ax $$ 5. Rearrange all terms to one side to set the equation to zero: $$ x^2 - 2xy + y^2 - 2ax = 0 $$ 6. This equation represents a conic section depending on the value of $a$. To analyze it further, you might solve for $y$ in terms of $x$ or vice versa, or consider specific values of $a$. 7. For example, solving for $y$: $$ x^2 - 2xy + y^2 = 2ax $$ $$ y^2 - 2xy + x^2 - 2ax = 0 $$ 8. Treating as a quadratic in $y$: $$ y^2 - 2x y + (x^2 - 2ax) = 0 $$ 9. Use the quadratic formula for $y$: $$ y = \frac{2x \pm \sqrt{(2x)^2 - 4 \cdot 1 \cdot (x^2 - 2ax)}}{2} $$ 10. Simplify under the square root: $$ (2x)^2 - 4(x^2 - 2ax) = 4x^2 - 4x^2 + 8ax = 8ax $$ 11. So, $$ y = \frac{2x \pm \sqrt{8ax}}{2} = x \pm \sqrt{2ax} $$ 12. This gives two branches of $y$ depending on $x$ and $a$. 13. Note that for real $y$, the expression under the square root must be non-negative: $$ 2ax \geq 0 $$ 14. This means $x$ and $a$ must have the same sign or one of them is zero. 15. Summary: The equation $ (x - y)^2 = 2ax $ can be rewritten as $$ y = x \pm \sqrt{2ax} $$ which describes two curves depending on the parameter $a$. Final answer: $$ y = x \pm \sqrt{2ax} $$