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)
Set up the distance equation based on the definition of a parabola: ( x − 4 ) 2 + ( y − 6 ) 2 = ∣ y + 6∣ .
Square both sides and expand: ( x − 4 ) 2 + y 2 − 12 y + 36 = y 2 + 12 y + 36 .
Simplify the equation by cancelling terms: ( x − 4 ) 2 = 24 y .
State the final equation of the parabola: ( x − 4 ) 2 = 24 y .
Explanation
Problem Analysis The problem provides the focus and directrix of a parabola and asks for its equation. We will use the definition of a parabola as the set of all points equidistant to 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 ( 4 , 6 ) is given by the distance formula: ( x − 4 ) 2 + ( y − 6 ) 2 The distance from ( x , y ) to the directrix y = − 6 is the perpendicular distance, which is: ∣ y − ( − 6 ) ∣ = ∣ y + 6∣
Equating Distances By the definition of a parabola, these two distances must be equal: ( x − 4 ) 2 + ( y − 6 ) 2 = ∣ y + 6∣
Squaring Both Sides Square both sides of the equation to eliminate the square root: ( x − 4 ) 2 + ( y − 6 ) 2 = ( y + 6 ) 2
Expanding Terms Expand the squared terms: ( x − 4 ) 2 + y 2 − 12 y + 36 = y 2 + 12 y + 36
Simplifying the Equation Simplify the equation by canceling out the y 2 and 36 terms: ( x − 4 ) 2 − 12 y = 12 y
Isolating the Squared Term Isolate the ( x − 4 ) 2 term: ( x − 4 ) 2 = 24 y
Final Equation The equation of the parabola is ( x − 4 ) 2 = 24 y .
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. Understanding the relationship between the focus, directrix, and equation of a parabola is crucial in optimizing the design for maximum efficiency.
