At what points does the line y = x intersect the unit circle defined by the equation x² + y² = 1?

To find the points where the line y = x intersects the unit circle described by the equation x² + y² = 1, we can follow these steps:

1. Substitute the equation of the line into the equation of the circle. Since y = x, we can replace y in the circle’s equation:

   x² + (x)² = 1

2. This simplifies to:

   2x² = 1

3. Now, divide both sides by 2:

   x² = 1/2

4. Taking the square root of both sides gives us:

   x = ±√(1/2) = ±1/√2 = ±√2/2  

5. Since y = x, the corresponding y values will be the same as the x values:

   y = ±√(1/2) = ±√2/2

Thus, the points of intersection between the line and the unit circle are:
(√2/2, √2/2) and (-√2/2, -√2/2).

These points nicely illustrate how a line can intersect a circle at two distinct locations, and in this case, they correspond to the angles of 45 degrees (π/4 radians) and 225 degrees (5π/4 radians) in polar coordinates, showcasing the symmetry of both geometric figures.