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Chapter 6: Problem 83

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

Short Answer

Expert verified
The statement 'It is possible for a parabola to intersect its directrix' is False.

Step by step solution

01

Recall Definition of a Parabola

The definition of a parabola states that it is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). This means the distance of any point on the parabola to the focus is equal to its distance to the directrix.
02

Recall Property of Parabola

A known property of a parabola is that it always 'opens' towards its focus and away from its directrix. Hence, in theory, the parabola won鈥檛 intersect the directrix due to this 'opening away' property.
03

Conclusion Regarding the Original Statement

By taking into account the definition and the properties of a parabola, it can be concluded that the statement 'It is possible for a parabola to intersect its directrix' is false because a parabola should never intersect its directrix according to its mathematical definition and properties.

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Most popular questions from this chapter

The equations of a parabola and a tangent line to the parabola are given. Use a graphing utility to graph both equations in the same viewing window. Determine the coordinates of the point of tangency. \(x^{2}+12 y=0 \quad x+y-3=0\)

An __________ is the set of all points \((x, y)\) in a plane, the sum of whose distances from two distinct fixed points, called _______ is constant.

Halley's comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately 0.967 . The length of the major axis of the orbit is approximately 35.88 astronomical units. (An astronomical unit is about 93 million miles.) (a) Find an equation of the orbit. Place the center of the orbit at the origin, and place the major axis on the \(x\) -axis. (b) Use a graphing utility to graph the equation of the orbit. (c) Find the greatest (aphelion) and smallest (perihelion) distances from the sun's center to the comet's center.

The path of a softball is modeled by \(-12.5(y-7.125)=(x-6.25)^{2},\) where the coordinates \(x\) and \(y\) are measured in feet, with \(x=0\) corresponding to the position from which the ball was thrown. (a) Use a graphing utility to graph the trajectory of the softball. (b) Use the trace feature of the graphing utility to approximate the highest point and the range of the trajectory.

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.
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