A parabola can be represented by the equation [tex]$y^2=12x$[/tex]. Which equation represents the directrix?
A. [tex]$y=-3$[/tex]
B. [tex]$y=3$[/tex]
C. [tex]$x=-3$[/tex]
D. [tex]$x=3$[/tex]
Answer (2)
We are given the equation of a parabola y 2 = 12 x .
We compare it to the standard form y 2 = 4 a x to find a , which represents the distance from the vertex to the directrix.
We determine that 4 a = 12 , so a = 3 .
Since the parabola opens to the right and the vertex is at the origin, the directrix is a vertical line x = − a , thus x = − 3 .
x = − 3
Explanation
Problem Analysis The equation of the parabola is given as y 2 = 12 x . We need to find the equation of the directrix.
General Form of Parabola The general form of a parabola opening to the right is y 2 = 4 a x , where a is the distance from the vertex to the focus and from the vertex to the directrix.
Finding the value of a Comparing the given equation y 2 = 12 x with the general form y 2 = 4 a x , we have 4 a = 12 . Dividing both sides by 4, we get a = 3 .
Determining the Directrix Since the vertex of the parabola y 2 = 12 x is at the origin (0, 0) and the parabola opens to the right, the directrix is a vertical line x = − a .
Equation of the Directrix Substituting the value of a = 3 into the equation x = − a , we find the equation of the directrix: x = − 3 .
Examples
Understanding parabolas and their directrices is crucial in various fields, such as satellite dish design. The directrix helps determine the shape and focus of the dish, ensuring optimal signal reception. Similarly, in optics, the directrix plays a key role in designing lenses that focus light rays efficiently. By grasping these concepts, students can apply them to real-world engineering and technological applications.
The equation of the directrix for the parabola y 2 = 12 x is determined to be x = − 3 . This is found by comparing the given equation to the standard form of a parabola and solving for the parameter a . Hence, the correct answer is option C: x = − 3 .
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