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 (2)
Define the parabola using the focus (0, -2) and directrix y = 2.
Calculate the distance from a point (x, y) on the parabola to the focus and to the directrix.
Equate the two distances and simplify the resulting equation.
Obtain the equation of the parabola: x 2 = − 8 y .
Explanation
Problem Analysis The problem provides the focus and directrix of a parabola and asks for its equation. The focus is at (0, -2) and the directrix is y = 2. We will use the definition of a parabola as the set of all points equidistant from the focus and the directrix to derive the equation.
Distance Calculations Let (x, y) be a point on the parabola. The distance from (x, y) to the focus (0, -2) is given by the distance formula: ( x − 0 ) 2 + ( y − ( − 2 ) ) 2 = x 2 + ( y + 2 ) 2 The distance from (x, y) to the directrix y = 2 is the perpendicular distance, which is |y - 2|.
Equating Distances According to the definition of a parabola, the distance from any point (x, y) on the parabola to the focus must equal the distance from that point to the directrix. Therefore, we set these two distances equal to each other: x 2 + ( y + 2 ) 2 = ∣ y − 2∣
Squaring Both Sides To eliminate the square root and absolute value, we square both sides of the equation: ( x 2 + ( y + 2 ) 2 ) 2 = ( ∣ y − 2∣ ) 2 x 2 + ( y + 2 ) 2 = ( y − 2 ) 2
Expanding Terms Now, we expand the squared terms: x 2 + ( y 2 + 4 y + 4 ) = ( y 2 − 4 y + 4 ) x 2 + y 2 + 4 y + 4 = y 2 − 4 y + 4
Simplifying the Equation Next, we simplify the equation by canceling out the y 2 and 4 terms from both sides: x 2 + 4 y = − 4 y
Combining y Terms Combine the y terms by adding 4y to both sides: x 2 = − 8 y
Final Equation Therefore, the equation of the parabola is: x 2 = − 8 y This matches one of the given options.
Examples
Parabolas are commonly used in the design of satellite dishes and reflecting telescopes. The parabolic shape focuses incoming signals or light to a single point (the focus), allowing for efficient signal reception or image formation. Understanding the relationship between the focus, directrix, and equation of a parabola is crucial in optimizing the design of these devices.
The equation of the parabola with a focus at (0, -2) and directrix y = 2 is derived as follows: Setting the distances from a point (x, y) to the focus and directrix equal leads to the final equation x 2 = − 8 y . Therefore, the answer is option D: x 2 = − 8 y .
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