Question

investigating the role of the focus and directrix
the graph shows a parabola whose focus is point f and directrix is the line d. move point p to several different locations on the graph.

1. what is the relationship between the lengths of segments pf and pq as point p moves along the graph?

options: pf < pq, pf = pq, pf > pq
(graph description: a parabola on a coordinate grid with focus f, directrix d, point p on the parabola, pq = 2.00 units, pf = 2.00 units)

Explanation:

A parabola is defined as the set of all points \( P \) in a plane such that the distance from \( P \) to the focus \( F \) is equal to the distance from \( P \) to the directrix \( d \). In the given graph, for point \( P \), \( PF \) is the distance from \( P \) to the focus \( F \), and \( PQ \) (where \( Q \) is the foot of the perpendicular from \( P \) to the directrix \( d \)) is the distance from \( P \) to the directrix. By the definition of a parabola, these two distances should be equal for any point \( P \) on the parabola. The example given in the graph also shows \( PF = 2.00 \) units and \( PQ = 2.00 \) units, which supports this relationship.

Answer:

\( \text{PF} = \text{PQ} \) (Option: \( \text{PF} = \text{PQ} \))