The focus of a parabola is located at $(4,0)$, and the directrix is located at $x=-4$. Which equation represents the parabola?
A. $y^2=-x$
B. $y^2=x$
C. $y^2=-16 x$
D. $y^2=16 x
Answer (2)
Define the parabola as the set of points equidistant from the focus ( 4 , 0 ) and the directrix x = − 4 .
Express the distance from a point ( x , y ) on the parabola to the focus and to the directrix.
Equate the two distances: ( x − 4 ) 2 + y 2 = ∣ x + 4∣ .
Simplify the equation to obtain the parabola's equation: y 2 = 16 x .
Explanation
Analyze the problem Let's analyze the given information. The focus of the parabola is at (4, 0), and the directrix is the line x = -4. We need to find the equation of the parabola.
Set up the distances Recall the definition of a parabola: it is the set of all points (x, y) that are equidistant to the focus and the directrix. The distance from a point (x, y) to the focus (4, 0) is given by the distance formula:
( x − 4 ) 2 + ( y − 0 ) 2
The distance from a point (x, y) to the directrix x = -4 is the perpendicular distance, which is:
∣ x − ( − 4 ) ∣ = ∣ x + 4∣
Equate the distances Now, set the distance to the focus equal to the distance to the directrix:
( x − 4 ) 2 + y 2 = ∣ x + 4∣
Square both sides Square both sides of the equation to eliminate the square root:
( x − 4 ) 2 + y 2 = ( x + 4 ) 2
Expand the equation Expand both sides of the equation:
x 2 − 8 x + 16 + y 2 = x 2 + 8 x + 16
Simplify the equation Simplify the equation by canceling out the x 2 and 16 terms:
− 8 x + y 2 = 8 x
Isolate y^2 Isolate y 2 to get the equation of the parabola:
y 2 = 16 x
State the final equation Therefore, the equation of the parabola is y 2 = 16 x .
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, where the receiver or detector is placed. This allows for efficient collection of signals or light from a large area. The same principle is used in reverse in spotlights and car headlights, where a light source at the focus projects a parallel beam of light.
The equation of the parabola with focus (4, 0) and directrix x = -4 is derived by equating the distances from any point (x, y) on the parabola to the focus and directrix. Through simplification, we find that the equation is y 2 = 16 x . Therefore, the correct answer is option D.
;
