A parabola is represented by the equation $y^2=5 x$. Which equation represents the directrix?
A. $y=-20$
B. $x=-20$
C. $y=-\frac{5}{4}$
D. $x=-\frac{5}{4}$
Answer (1)
Identify the given equation of the parabola: y 2 = 5 x .
Recognize the general form of a parabola opening to the right: y 2 = 4 a x .
Determine the value of a by comparing the given equation with the general form: a = 4 5 .
State the equation of the directrix for a parabola opening to the right with vertex at the origin: x = − a . Substitute the value of a to find the directrix: x = − 4 5 .
Explanation
Problem Analysis We are given the equation of a parabola as y 2 = 5 x . Our goal is to find the equation of the directrix.
General Form of a Parabola The general form of a parabola opening to the right is given by y 2 = 4 a x , where a is the distance from the vertex to the focus and also the distance from the vertex to the directrix. The vertex of this parabola is at the origin (0,0).
Finding the Value of a Comparing the given equation y 2 = 5 x with the general form y 2 = 4 a x , we can find the value of a . We have 4 a = 5 , so solving for a gives us a = 4 5 .
Equation of the Directrix Since the parabola opens to the right and its vertex is at the origin, the directrix is a vertical line given by the equation x = − a .
Substituting the Value of a Substituting the value of a = 4 5 into the equation x = − a , we get the equation of the directrix as x = − 4 5 .
Examples
Understanding parabolas and their directrices is crucial in various fields, such as designing satellite dishes or understanding the trajectory of projectiles. For instance, if you're designing a satellite dish, the directrix helps determine the optimal placement of the receiver to capture signals efficiently. The reflective property of the parabola ensures that signals coming from a distant source are focused at the focus, and the directrix plays a key role in this property.
