A parabola has a vertex at the origin. The equation of the directrix of the parabola is [tex]$y=3$[/tex]. What are the coordinates of its focus?
A. [tex]$(0,3)$[/tex]
B. [tex]$(3,0)$[/tex]
C. [tex]$(0,-3)$[/tex]
D. [tex]$(-3,0)$[/tex]
Answer (2)
The vertex of the parabola is at the origin (0, 0).
The directrix is y = 3 .
The focus is equidistant from the vertex as the directrix, but on the opposite side.
The coordinates of the focus are ( 0 , − 3 ) .
Explanation
Analyze the problem The vertex of the parabola is at the origin (0, 0), and the directrix is given by the equation y = 3 . We need to find the coordinates of the focus.
Determine the axis of symmetry The vertex of a parabola is equidistant from the focus and the directrix. Since the vertex is at (0, 0) and the directrix is y = 3 , the focus must lie on the y-axis.
Calculate the distance The distance between the vertex and the directrix is ∣3 − 0∣ = 3 . Therefore, the focus must be 3 units away from the vertex in the opposite direction of the directrix.
Find the coordinates of the focus Since the directrix is above the vertex, the focus must be below the vertex. Thus, the coordinates of the focus are (0, -3).
State the final answer The coordinates of the focus are ( 0 , − 3 ) .
Examples
Parabolas are commonly used in the design of satellite dishes and reflecting telescopes. The focus of a parabolic reflector is the point where incoming parallel rays (like radio waves or light) converge after being reflected by the parabolic surface. Knowing the location of the focus is crucial for placing the receiver or detector to efficiently collect the signal or light. In this case, if you have a parabolic dish with its vertex at the origin and a directrix at y=3, you would place the receiver at the focus, which is at (0, -3).
The coordinates of the focus of the parabola are (0, -3). This point lies 3 units below the vertex at the origin, opposite to the directrix located at y = 3. The correct answer is C. (0, -3).
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