Chapter 8.1, Problem 71E

Solution

a.

To graph: Placing a line (directrix) and a point (focus) not on the line in the construction window

Given information:

Line

Point (focus)

Graph:

  

Interpretation:

Drew a random line for the directrix and a point not on the line.

b.

To graph: Construct a point on the directrix, and then the segment .

Given information:

Point .

Segment .

Graph:

  

Interpretation:

Place a point on line , naming it and connect and ,forming the segment .

c.

To graph: Construct a point where the perpendicular bisector of meets the line perpendicular to through .

Given information:

Point .

The perpendicular bisector of .

Graph:

  

Interpretation:

Construct the perpendicular bisector of by locating its midpoint first then constructing the perpendicular line through it. Then, construct the perpendicular line to line through . The point of intersection of the two lines constructed is point ..

d.

To graph: What curve does trace out as moves.

Given information:

Point .

Move point .

Graph:

  

Interpretation:

Select point and use the Trace Intersection feature (from Display). Move point along line back and forth and you will notice that point traces out a parabola as shown.

e.

To prove: Prove that your answer parabola is correct.

Given information:

Parabola equation trace.

Proof:

With labels as shown, can express the coordinates of using the point-slope equation of the line $P M$

$ $

This is the equation of a parabola with vertex at and focus at where .