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)
Compare the given equation y 2 = − x with the standard form y 2 = 4 a x .
Find the value of a by equating 4 a = − 1 , which gives a = − 4 1 .
Determine the focus using the formula ( a , 0 ) , resulting in focus at ( − 4 1 , 0 ) .
Determine the directrix using the formula x = − a , resulting in directrix x = 4 1 .
focus: ( − 4 1 , 0 ) ; directrix: x = 4 1
Explanation
Problem Analysis The equation of the parabola is given as y 2 = − x . We need to find the coordinates of the focus and the equation of the directrix.
Finding the value of a The standard form of a parabola opening to the left is y 2 = 4 a x , where a < 0 . Comparing this with the given equation y 2 = − x , we can write − x = 4 a x . Thus, 4 a = − 1 , which gives us a = − 4 1 .
Focus and Directrix Formulas For a parabola in the form y 2 = 4 a x , the focus is located at ( a , 0 ) and the directrix is given by the equation x = − a .
Finding the Focus Substituting a = − 4 1 into the coordinates of the focus ( a , 0 ) , we get the focus at ( − 4 1 , 0 ) .
Finding the Directrix Substituting a = − 4 1 into the equation of the directrix x = − a , we get x = − ( − 4 1 ) , which simplifies to 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 onto a single point, the focus. The location of the focus determines where the receiver should be placed to optimally capture the signal. Similarly, the path of a projectile, neglecting air resistance, follows a parabolic trajectory, and knowing the focus and directrix can help predict its range and maximum height.
