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Chapter 6: Problem 78

Determine whether the statement is true or false. Justify your answer. If the vertex and focus of a parabola are on a horizontal line, then the directrix of the parabola is vertical.

Short Answer

Expert verified
The statement is true. If the vertex and focus of a parabola are on a horizontal line, then the directrix of the parabola is indeed a vertical line.

Step by step solution

01

Conceptual Understanding

By definition, the vertex is the highest or lowest point of a parabola, depending on whether the parabola opens upwards or downwards. The focus is a special point inside the parabola from which all points on the parabola are equidistant. There is a line called the directrix in line with the vertex and focus but on the exterior of the parabola. For any point on the parabola, the distance to the focus is equal to the distance to the directrix.
02

Analyzing the Statement

When the vertex and focus of a parabola are on a horizontal line, it means the parabola could be opening to the right (if the focus is to the right of the vertex) or to the left (if the focus is to the left of the vertex). No matter which direction it opens, the directrix will always be a vertical line as it must be perpendicular to the axis of symmetry (the line that passes through the focus and vertex).
03

Conclusion Based on the Analysis

Since the directrix is perpendicular to the line drawn between the vertex and the focus (which is a horizontal line in this case) then the directrix has to be a vertical line. This is always the case irrespective of whether the parabola opens to the left or the right.

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