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 standard form of a parabola opening to the right: y 2 = 4 a x .
Determine the value of a by equating 4 a = 5 , which gives a = 4 5 .
State the equation of the directrix as x = − a , and substitute the value of a to get the final answer: x = − 4 5 .
Explanation
Analyze the given equation We are given the equation of a parabola as y 2 = 5 x . Our goal is to find the equation of the directrix. Let's first analyze the given equation and recall the standard form of a parabola to determine the directrix.
Recall the standard form The standard 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).
Find the value of a Comparing the given equation y 2 = 5 x with the standard form y 2 = 4 a x , we can equate the coefficients to find the value of a :
4 a = 5
Solving for a , we get:
a = 4 5
Determine the 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 we found:
x = − 4 5
State the final answer Therefore, the equation of the directrix is x = − 4 5 .
Examples
Understanding parabolas and their directrices is crucial in various fields, such as antenna design. For instance, satellite dishes are shaped like paraboloids, and the receiver is placed at the focus. The directrix helps define the shape of the parabola, ensuring signals are properly focused. This principle is also used in designing reflective telescopes and solar cookers, where precise focusing is essential for optimal performance.
