A parabola has the focus at $(4,6)$ and the directrix $y=-6$. Which equation represents this parabola?
$(x-4)^2=24 y$
$(x-4)^2=\frac{1}{24} y$
$(x-4)^2=-24 y$
$(x-4)^2=-\frac{1}{24} y
Answer (1)
Define the parabola as the set of points equidistant from the focus ( 4 , 6 ) and the directrix y = − 6 .
Express the distance from a point ( x , y ) on the parabola to the focus and to the directrix.
Equate the two distances and simplify the equation.
Obtain the equation of the parabola: ( x − 4 ) 2 = 24 y .
Explanation
Problem Analysis The focus of the parabola is at the point ( 4 , 6 ) . The directrix of the parabola is the line y = − 6 . We want to find the equation of the parabola.
Definition of Parabola Recall the definition of a parabola: the set of all points equidistant to the focus and the directrix. Let ( x , y ) be a point on the parabola. The distance from ( x , y ) to the focus ( 4 , 6 ) is ( x − 4 ) 2 + ( y − 6 ) 2 . The distance from ( x , y ) to the directrix y = − 6 is ∣ y − ( − 6 ) ∣ = ∣ y + 6∣ .
Equating Distances Set these two distances equal to each other: ( x − 4 ) 2 + ( y − 6 ) 2 = ∣ y + 6∣ . Square both sides of the equation to eliminate the square root: ( x − 4 ) 2 + ( y − 6 ) 2 = ( y + 6 ) 2 .
Expanding and Simplifying Expand the squared terms: ( x − 4 ) 2 + y 2 − 12 y + 36 = y 2 + 12 y + 36 . Simplify the equation by canceling out the y 2 and 36 terms: ( x − 4 ) 2 − 12 y = 12 y .
Isolating the Term Isolate the ( x − 4 ) 2 term: ( x − 4 ) 2 = 24 y . Thus, the equation of the parabola is ( x − 4 ) 2 = 24 y .
Examples
Parabolas are commonly seen in the real world, such as the trajectory of a ball thrown in the air or the shape of satellite dishes. Understanding the equation of a parabola allows us to model and analyze these phenomena. For example, if we know the focus and directrix of a satellite dish, we can determine its shape using the equation of a parabola, ensuring that it efficiently collects signals at the focus.
