A parabola can be represented by the equation [tex]$x^2=-ny$[/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)
The given equation is x 2 = − n y .
Rewrite the equation in the standard form x 2 = 4 p y , where p = − 4 n .
The focus of the parabola is at ( 0 , p ) , so the focus is ( 0 , − 4 n ) .
Since 0"> n > 0 , the y -coordinate of the focus is negative, and the correct answer is ( 0 , − 5 ) .
Explanation
Analyze the problem We are given the equation of a parabola as x 2 = − n y , where n is a positive constant. We need to find the coordinates of the focus of this parabola.
Rewrite the equation The standard form of a parabola opening downwards is x 2 = 4 p y , where the focus is at ( 0 , p ) . Comparing the given equation x 2 = − n y with the standard form, we can write 4 p = − n .
Determine the focus Solving for p , we get p = − 4 n . Since the focus of the parabola is at ( 0 , p ) , the coordinates of the focus are ( 0 , − 4 n ) .
Check answer choices Since n is positive, − 4 n is negative. Now we check the given answer choices to see which one matches our result. The answer choices are:
( − 5 , 0 )
( 5 , 0 )
( 0 , 5 )
( 0 , − 5 )
Since the focus is of the form ( 0 , − 4 n ) , we can see that the x -coordinate must be 0, and the y -coordinate must be negative. The only option that satisfies this condition is ( 0 , − 5 ) .
Confirm the answer If the focus is ( 0 , − 5 ) , then − 4 n = − 5 , which means n = 20 . So the equation of the parabola is x 2 = − 20 y . This confirms that the focus is indeed at ( 0 , − 5 ) .
Examples
Parabolas are commonly used in the design of satellite dishes and reflecting telescopes. The focus of the parabola is the point where incoming parallel rays converge after reflection. Knowing the equation of a parabolic reflector, we can determine the precise location of the focus to optimize signal reception or light collection. For example, if a satellite dish has a cross-section described by x 2 = − 20 y , the receiver should be placed at the focus, which we found to be ( 0 , − 5 ) . This ensures that the satellite signal is optimally concentrated at the receiver.
