Find the coordinates of the points where the line $y = x$ intersects the unit circle.

We start with the unit circle equation:

$x^2 + y^2 = 1$

Substituting $y = x$, we get:

$x^2 + x^2 = 1$

$2x^2 = 1$

$x^2 = \frac{1}{2}$

$x = \pm \frac{\sqrt{2}}{2}$

Since $y = x$, the coordinates are:

$(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

and

$( – \frac{\sqrt{2}}{2}, – \frac{\sqrt{2}}{2})$

We use the unit circle equation:

$x^2 + y^2 = 1$

Given $y = x$, substitute into the equation:

$x^2 + x^2 = 1$

$2x^2 = 1$

$x^2 = frac{1}{2}$

$x = pm frac{sqrt{2}}{2}$

Thus, the points of intersection are:

$(frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$

and

$( – frac{sqrt{2}}{2}, – frac{sqrt{2}}{2})$

Using the unit circle equation:

$x^2 + y^2 = 1$

Substitute $y = x$:

$2x^2 = 1$

$x = pm frac{sqrt{2}}{2}$

Coordinates are $(frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$ and $( – frac{sqrt{2}}{2}, – frac{sqrt{2}}{2})$

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