The focus of a parabola is located at $(0,-2)$. The directrix of the parabola is represented by $y=2$. Which equation represents the parabola?
A. $y^2=-2 x$
B. $x^2=-2 y$
C. $y^2=-8 x$
D. $x^2=-8 y$
Answer (1)
Find the vertex of the parabola as the midpoint between the focus and directrix.
Calculate the distance p between the vertex and the focus.
Use the vertex and p to form the equation of the parabola, considering it opens downwards.
The equation of the parabola is x 2 = − 8 y .
Explanation
Problem Analysis The focus of the parabola is at ( 0 , − 2 ) and the directrix is y = 2 . We need to find the equation of the parabola.
Finding the Vertex The vertex of a parabola is located exactly in the middle between the focus and the directrix. Since the focus is at ( 0 , − 2 ) and the directrix is y = 2 , the vertex will have an x -coordinate of 0. The y -coordinate will be the average of − 2 and 2 , which is 2 − 2 + 2 = 0 . Therefore, the vertex is at ( 0 , 0 ) .
Finding the Distance p The distance p between the vertex and the focus (or the vertex and the directrix) is the same. In this case, the distance between ( 0 , 0 ) and ( 0 , − 2 ) is ∣ − 2 − 0∣ = 2 . So, p = 2 .
Finding the Equation of the Parabola Since the focus is below the directrix, the parabola opens downward. The general equation for a parabola that opens downward with a vertex at ( h , k ) is ( x − h ) 2 = − 4 p ( y − k ) . In our case, the vertex is ( 0 , 0 ) , so h = 0 and k = 0 . Also, p = 2 . Substituting these values into the equation, we get ( x − 0 ) 2 = − 4 ( 2 ) ( y − 0 ) , which simplifies to x 2 = − 8 y .
Final Answer The equation of the parabola is x 2 = − 8 y .
Examples
Parabolas are commonly found in the design of satellite dishes and reflecting telescopes. The parabolic shape helps to focus incoming signals or light to a single point, the focus, where the receiver or detector is placed. Understanding the relationship between the focus, directrix, and equation of a parabola is crucial in optimizing the design for maximum efficiency.
