Show that the line y = x - 7 does not meet the circle (x + 2)^2 + y^2 = 33.

To find potential points of intersection between the line and the circle, we need to solve the equations simultaneously. So, we substitute y = x - 7 into the equation of the circle: (x + 2)² + y² = 33 (x + 2)² + (x - 7)² = 33 x² + 4x + 4 + x² - 14x + 49 = 33 (expand the brackets) 2x² - 10x + 20 = 0 (collect like terms) x² - 5x + 10 = 0 (divide each term by 2) Now, the discriminant b² - 4ac = (-5)² - 4 * 1 *10 = 25 - 40 = -15. As b² - 4ac < 0, there is no solution to the quadratic equation. So, the line does not meet the circle.