A parabola can be represented by the equation [tex]$x^2=-$[/tex] [tex]$20 y$[/tex]. What are the coordinates of the focus of the parabola?
A. [tex]$(-5,0)$[/tex]
B. [tex]$(5,0)$[/tex]
C. [tex]$(0,5)$[/tex]
D. [tex]$(0,-5)$[/tex]
Answer (1)
Identify the given equation of the parabola as x 2 = − 20 y .
Compare the given equation with the standard form x 2 = − 4 a y to determine the value of a .
Solve for a to find a = 5 .
Determine the coordinates of the focus as ( 0 , − a ) , which gives the final answer: ( 0 , − 5 ) .
Explanation
Analyze the given equation The given equation of the parabola is x 2 = − 20 y . We need to find the coordinates of the focus of this parabola. The standard form of a parabola that opens upwards or downwards is x 2 = 4 a y or x 2 = − 4 a y , where the focus is at ( 0 , a ) or ( 0 , − a ) , respectively. In our case, the parabola opens downwards because the coefficient of y is negative.
Compare with the standard form Comparing the given equation x 2 = − 20 y with the standard form x 2 = − 4 a y , we can write: − 4 a = − 20
Solve for a Now, we solve for a :
a = − 4 − 20 = 5
Determine the focus coordinates Since the equation is in the form x 2 = − 4 a y , the focus of the parabola is at ( 0 , − a ) . Substituting a = 5 , we get the coordinates of the focus as ( 0 , − 5 ) .
State the final answer Therefore, the coordinates of the focus of the parabola x 2 = − 20 y are ( 0 , − 5 ) .
Examples
Understanding parabolas is crucial in various real-world applications. For instance, the reflective properties of parabolic mirrors are used in satellite dishes and solar cookers to focus incoming signals or sunlight onto a single point, enhancing efficiency. Similarly, the trajectory of a projectile, like a ball thrown in the air, approximately follows a parabolic path, which is essential in sports and military applications for accurate targeting and prediction.
