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Chapter 6: Problem 84

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 false. When the vertex and focus of a parabola are on a horizontal line, the directrix is horizontal, not vertical.

Step by step solution

01

Understand the basics of a parabola

A parabola is defined as the set of all points that are equidistant from a point (the focus) and a line (the directrix). The vertex of the parabola is the point on the plane of the parabola that is equidistant from the focus and the directrix.
02

Consider a parabola with its vertex and focus on a horizontal line

Here, if we imagine or sketch a parabola with its focus and vertex on a horizontal line, we will see that the parabola opens either upwards or downwards. This visualization brings to clarity that the directrix, our line in question, must be a horizontal line as well for the points on the parabola to maintain equal distance with the focus and the directrix.
03

Make a conclusion

So, visualizing it or recalling the standard forms of a parabola, when the vertex and focus of a parabola are on a horizontal line, the parabola opens vertically, which means the directrix is horizontal. Therefore, the given statement is false.

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Most popular questions from this chapter

The receiver in a parabolic satellite dish is 4.5 feet from the vertex and is located at the focus (see figure). Write an equation for a cross section of the reflector. (Assume that the dish is directed upward and the vertex is at the origin.)

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Determine whether the statement is true or false. Justify your answer. It is possible for a parabola to intersect its directrix.
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