Question

Find the points of intersection of the graphs of the equations. $$ \begin{array}{l} \theta=\frac{\pi}{4} \\ r=2 \end{array} $$

Short answer

The points of intersection of the graphs are (1, 1) and (-1, -1).

Step-by-step solution

Convert Equations to Cartesian Form

Remember, polar coordinates (r, θ) can be converted to Cartesian coordinates (x, y) using the formulas \( x = r \cos{\theta} \) and \( y = r \sin{\theta} \). Convert \( \theta = \frac{\pi}{4} \) to the line equation \( y = x \) and \( r = 2 \) to the circle equation \( x^2 + y^2 = 4 \).

Solve for Intersection Points

The intersection points of the two graphs are solutions of both equations. Substitute \( y = x \) into the circle equation to get \( x^2 + x^2 = 4 \). This simplifies to \( 2x^2 = 4 \), from which we can solve \( x = \pm 1 \). Substituting \( x = 1 \) and \( x = -1 \) into \( y = x \) gives the corresponding \( y \) coordinates, so the intersection points are \( (1, 1) \) and \( (-1, -1) \).