If the directrix of a parabola is the horizontal line [tex]$y=3$[/tex], what is true of the parabola?
A. The focus is at [tex]$(0,3)$[/tex], and the equation for the parabola is [tex]$y^2=12 x$[/tex].
B. The focus is at [tex]$(0,-3)$[/tex], and the equation for the parabola is [tex]$x^2=-12 y$[/tex].
C. The focus is at [tex]$(3,0)$[/tex], and the equation for the parabola is [tex]$x^2=12 y$[/tex].
D. The focus is at [tex]$(-3,0)$[/tex], and the equation for the parabola is [tex]$y^2=-12 x$[/tex].
Answer (2)
Analyze each option by finding the vertex and the distance between the vertex and the focus.
Determine the equation of the parabola based on the focus and directrix.
Compare the derived equation with the equation provided in the option.
The correct option is: The focus is at ( 0 , − 3 ) , and the equation for the parabola is x 2 = − 12 y .
Explanation
Problem Analysis The directrix of the parabola is the horizontal line y = 3 . We need to determine which of the given options is correct.
Recall that a parabola is the set of all points equidistant from the focus and the directrix. The vertex of the parabola is the midpoint between the focus and the directrix. The distance from the vertex to the focus is denoted by p .
Analyze each option Let's analyze each option:
Option 1: The focus is at ( 0 , 3 ) , and the equation for the parabola is y 2 = 12 x .
Since the directrix is y = 3 and the focus is ( 0 , 3 ) , the focus lies on the directrix. This means we don't have a parabola, so this option is incorrect.
Option 2: The focus is at ( 0 , − 3 ) , and the equation for the parabola is x 2 = − 12 y .
The vertex is the midpoint between the focus ( 0 , − 3 ) and the directrix y = 3 . The coordinates of the vertex are ( 0 , 2 3 + ( − 3 ) ) = ( 0 , 0 ) .
The distance p between the vertex ( 0 , 0 ) and the focus ( 0 , − 3 ) is p = ∣ − 3 − 0∣ = 3 .
Since the directrix is y = 3 and the focus is below the directrix, the parabola opens downwards. The equation of the parabola is of the form ( x − h ) 2 = − 4 p ( y − k ) , where ( h , k ) is the vertex. In this case, ( h , k ) = ( 0 , 0 ) , so the equation is x 2 = − 4 ( 3 ) y , which simplifies to x 2 = − 12 y . This matches the given equation, so this option is correct.
Option 3: The focus is at ( 3 , 0 ) , and the equation for the parabola is x 2 = 12 y .
The vertex is the midpoint between the focus ( 3 , 0 ) and the directrix y = 3 . The coordinates of the vertex are ( 2 3 + 3 , 2 0 + 3 ) = ( 3 , 2 3 ) .
The distance p between the vertex ( 3 , 2 3 ) and the directrix y = 3 is p = ∣3 − 2 3 ∣ = 2 3 .
Since the directrix is above the focus, the parabola opens downwards. The equation of the parabola is of the form ( x − h ) 2 = − 4 p ( y − k ) , where ( h , k ) is the vertex. In this case, ( h , k ) = ( 3 , 2 3 ) , so the equation is ( x − 3 ) 2 = − 4 ( 2 3 ) ( y − 2 3 ) , which simplifies to ( x − 3 ) 2 = − 6 ( y − 2 3 ) . This does not match the given equation, so this option is incorrect.
Option 4: The focus is at ( − 3 , 0 ) , and the equation for the parabola is y 2 = − 12 x .
The vertex is the midpoint between the focus ( − 3 , 0 ) and the directrix y = 3 . The coordinates of the vertex are ( 2 − 3 − 3 , 2 0 + 3 ) = ( − 3 , 2 3 ) .
The distance p between the vertex ( − 3 , 2 3 ) and the directrix y = 3 is p = ∣3 − 2 3 ∣ = 2 3 .
Since the directrix is above the focus, the parabola opens downwards. The equation of the parabola is of the form ( x − h ) 2 = − 4 p ( y − k ) , where ( h , k ) is the vertex. In this case, ( h , k ) = ( − 3 , 2 3 ) , so the equation is ( x + 3 ) 2 = − 4 ( 2 3 ) ( y − 2 3 ) , which simplifies to ( x + 3 ) 2 = − 6 ( y − 2 3 ) . This does not match the given equation, so this option is incorrect.
Final Answer The correct option is the second one. The focus is at ( 0 , − 3 ) , and the equation for the parabola is x 2 = − 12 y .
Examples
Parabolas are commonly used in designing satellite dishes. The dish is shaped like a parabola, and the receiver is placed at the focus. This design ensures that all incoming signals are reflected to the receiver, maximizing signal strength. Understanding the relationship between the focus, directrix, and equation of a parabola is crucial for optimizing the design of such devices.
The correct option is B: The focus is at (0,-3), and the equation is x^2 = -12y. This matches the requirements for a parabola with a directrix at y = 3. The vertex is at (0,0), confirming the calculations.
;
