A parabola can be represented by the equation $y^2=-x$. What are the coordinates of the focus and the equation of the directrix?
A. focus: $\left(-\frac{1}{4}, 0\right)$; directrix: $x=\frac{1}{4}$
B. focus: $\left{\frac{1}{4}, 0\right\}$ ) directrix: $x=-\frac{1}{4}$
C. focus: $(-4,0)$; directrix: $x=4$
D. focus: $(4,0)$; directrix: $x=-4$
Answer (1)
Identify the given equation of the parabola as y 2 = − x .
Recognize the standard form of a left-opening parabola as y 2 = 4 p x and equate 4 p = − 1 to find p = − 4 1 .
Determine the focus coordinates using p , which are ( − 4 1 , 0 ) .
Find the equation of the directrix using x = − p , resulting in x = 4 1 .
focus: ( − 4 1 , 0 ) ; directrix: x = 4 1
Explanation
Problem Analysis We are given the equation of a parabola as y 2 = − x . Our goal is to find the coordinates of the focus and the equation of the directrix.
Standard Form of Parabola The standard form of a parabola opening to the left is given by y 2 = 4 p x , where p < 0 . The focus of such a parabola is at ( p , 0 ) , and the equation of the directrix is x = − p . We need to compare the given equation with the standard form to find the value of p .
Finding the Value of p Comparing y 2 = − x with y 2 = 4 p x , we can write 4 p = − 1 . Solving for p , we get:
4 p = − 1 p = 4 − 1
So, p = − 4 1 .
Finding the Focus Now that we have the value of p , we can find the coordinates of the focus. The focus is at ( p , 0 ) , so the focus is at ( − 4 1 , 0 ) .
Finding the Directrix The equation of the directrix is x = − p . Substituting the value of p , we get:
x = − ( − 4 1 ) x = 4 1
So, the equation of the directrix is x = 4 1 .
Final Answer Therefore, the coordinates of the focus are ( − 4 1 , 0 ) , and the equation of the directrix is x = 4 1 .
Examples
Understanding parabolas is crucial in various fields like physics and engineering. For instance, satellite dishes and radio telescopes use parabolic reflectors to focus incoming signals to a single point, the focus. The location of the focus determines where the receiver should be placed to optimally capture the signal. Similarly, the design of car headlights utilizes parabolic reflectors to project light in a parallel beam, enhancing visibility. Knowing the equation of the parabola allows engineers to precisely calculate the focus and design efficient and effective optical systems.
