At what points will the line y = x intersect the unit circle x² + y² = 1?

To find the intersection points between the line defined by the equation y = x and the unit circle defined by x² + y² = 1, we can substitute the line’s equation into the circle’s equation.

Substituting y = x into x² + y² = 1 gives us:

x² + (x)² = 1

This simplifies to:

2x² = 1

Now, dividing both sides by 2:

x² = 1/2

Taking the square root of both sides results in two possible values for x:

x = ±√(1/2) = ±1/√2 = ±√2/2

Now that we have the values of x, we can find the corresponding y values using the original line equation y = x:
When x = √2/2, y = √2/2.
When x = -√2/2, y = -√2/2.

Thus, the intersection points of the line y = x with the unit circle x² + y² = 1 are:

• Point 1: (√2/2, √2/2)
• Point 2: (-√2/2, -√2/2)