This question tests AP Precalculus skills: understanding parametrically defined circles and lines (focus on algebraic and geometric interpretation). Parametric equations represent geometric figures by defining coordinates as functions of a parameter, often time (t) or distance (r). For this question, the circle has center (-2,1) and radius 6, while the line passes through (4,1) with slope 2. Choice A is correct because substituting x=4+r and y=1+2r into (x+2)²+(y-1)²=36 gives r=0 and r=-4, yielding intersection points (4,1) and (0,-7). Choice C is incorrect as it finds one correct point but miscalculates the second intersection by using the wrong parameter value. To help students: Practice solving quadratic equations that arise from circle-line intersections. Verify solutions by checking that both points satisfy the original circle equation.

This question tests AP Precalculus skills: understanding parametrically defined circles and lines (focus on algebraic and geometric interpretation). Parametric equations represent geometric figures by defining coordinates as functions of a parameter, often time (t) or distance (u). For this question, the circle has center (1,2) and radius 4, while the line passes through (5,6) with slope -1. Choice A is correct because substituting x=5+u and y=6-u into the circle equation (x-1)²+(y-2)²=16 gives u=0 and u=4, yielding intersection points (5,6) and (9,2). Choice C is incorrect due to an algebraic error in solving the quadratic equation, resulting in wrong parameter values. To help students: Emphasize careful algebraic manipulation when substituting parametric equations. Practice converting between parametric and Cartesian forms, and verify solutions by substituting back into both original equations.

This question tests AP Precalculus skills: understanding parametrically defined circles and lines (focus on algebraic and geometric interpretation). Parametric equations represent geometric figures by defining coordinates as functions of a parameter, often time (t) or distance (k). For this question, the circle has center (-1,4) and radius 3, while the line passes through (2,4) with slope 2. Choice A is correct because substituting x=2+k and y=4+2k into (x+1)²+(y-4)²=9 gives k=0 and k=-2, yielding intersection points (2,4) and (0,0). Choice C is incorrect as it finds one correct point but uses the wrong parameter value for the second intersection, resulting in a point outside the circle. To help students: Develop systematic approaches to solving circle-line intersection problems. Practice interpreting parameter values geometrically to understand what they represent on the line.

This question tests AP Precalculus skills: understanding parametrically defined circles and lines (focus on algebraic and geometric interpretation). Parametric equations represent geometric figures by defining coordinates as functions of a parameter, often time (t) or distance (p). For this question, the circle has center (0,2) and radius 5, while the line passes through (5,2) with slope 1. Choice A is correct because substituting x=5+p and y=2+p into x²+(y-2)²=25 gives p=0 and p=-5, yielding intersection points (5,2) and (0,-3). Choice B is incorrect as it miscalculates the y-coordinate of the first intersection point, likely due to a sign error when evaluating the parametric equation. To help students: Carefully track signs when substituting parameter values back into parametric equations. Verify that calculated points satisfy both the circle and line equations.