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 (1)
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
Problem Analysis The focus of the parabola is at ( 4 , 0 ) , and the directrix is the line x = − 4 . We want to find the equation of the parabola.
Definition of Parabola A parabola is defined as the set of all points equidistant from the focus and the directrix. Let ( x , y ) be a point on the parabola. The distance from ( x , y ) to the focus ( 4 , 0 ) is ( x − 4 ) 2 + ( y − 0 ) 2 . The distance from ( x , y ) to the directrix x = − 4 is ∣ x − ( − 4 ) ∣ = ∣ x + 4∣ .
Equating Distances Setting the two distances equal to each other, we have: ( x − 4 ) 2 + y 2 = ∣ x + 4∣ Squaring both sides of the equation to eliminate the square root and absolute value, we get: ( x − 4 ) 2 + y 2 = ( x + 4 ) 2
Expanding the Equation Expanding both sides: x 2 − 8 x + 16 + y 2 = x 2 + 8 x + 16
Simplifying the Equation Simplifying the equation by canceling out the x 2 and 16 terms: y 2 − 8 x = 8 x Isolating y 2 to obtain the equation of the parabola: y 2 = 16 x
Final Answer The equation of the parabola is y 2 = 16 x .
Examples
Parabolas are commonly seen in the real world, such as in the shape of satellite dishes or the path of a projectile. Understanding the equation of a parabola allows engineers to design these structures efficiently. For example, knowing the focus and directrix helps in optimizing the signal reception of a satellite dish.
