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 standard form of a parabola opening to the left: y 2 = 4 p x .
Compare the given equation y 2 = − x with the standard form to find 4 p = − 1 , and solve for p : p = − 4 1 .
Determine the focus using the formula ( p , 0 ) , which gives ( − 4 1 , 0 ) .
Determine the directrix using the formula x = − p , which gives x = 4 1 .
The focus is ( − 4 1 , 0 ) and the directrix is 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.
Identifying the Standard Form The standard form of a parabola that opens to the left is given by y 2 = 4 p x , where the focus is at ( p , 0 ) and the equation of the directrix is x = − p . We need to compare the given equation with this standard form to find the value of p .
Solving for p Comparing y 2 = − x with y 2 = 4 p x , we can see that 4 p = − 1 . Now, we solve for p :
4 p = − 1 p = 4 − 1 So, p = − 4 1 .
Finding the Focus and Directrix Now that we have the value of p , we can find the coordinates of the focus and the equation of the directrix.
The focus is at ( p , 0 ) , which is ( − 4 1 , 0 ) .
The directrix is given by x = − p , which is x = -\left(-\frac{1}{4}\\right) , so 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, such as satellite dish design. The reflective property of a parabola ensures that signals from a satellite are focused at a single point, the focus, where the receiver is placed. The directrix helps define the shape of the parabola, ensuring optimal signal collection. By knowing the equation of a parabola, engineers can accurately determine the placement of the receiver and the shape of the dish for maximum efficiency.
