A parabola, with its vertex at the origin, has a directrix at $y=3$.
Which statements about the parabola are true? Select two options.
A. The focus is located at $(0,-3)$.
B. The parabola opens to the left.
C. The $p$ value can be determined by computing 4(3).
D. The parabola can be represented by the equation $x^2=-12 y$.
E. The parabola can be represented by the equation $y^2=12 x$.
Answer (2)
The parabola opens downwards because the directrix y = 3 is above the vertex ( 0 , 0 ) .
The focus is at ( 0 , − 3 ) since it's equidistant from the vertex as the directrix.
The value of p is 3, representing the distance from the vertex to the focus.
The equation of the parabola is x 2 = − 12 y , derived from the standard form x 2 = − 4 p y with p = 3 .
Therefore, the two correct statements are:
The focus is located at ( 0 , − 3 ) .
The parabola can be represented by the equation x 2 = − 12 y .
x 2 = − 12 y
Explanation
Analyze the Parabola's Orientation The vertex of the parabola is at the origin (0,0), and the directrix is at y = 3 . Since the directrix is a horizontal line, the parabola opens either upwards or downwards. Because the directrix is above the vertex, the parabola opens downwards.
Determine the Focus The focus of a parabola is equidistant from the vertex as the directrix. Since the directrix is at y = 3 , the distance from the vertex (0,0) to the directrix is 3. Therefore, the focus is at ( 0 , − 3 ) .
Find the Value of p The distance between the vertex and the focus (or the vertex and the directrix) is the value of p . In this case, p = 3 .
Write the Equation of the Parabola The general equation for a parabola that opens downwards with a vertex at the origin is x 2 = − 4 p y . Substituting p = 3 into the equation, we get x 2 = − 4 ( 3 ) y , which simplifies to x 2 = − 12 y .
Evaluate the Statements Now, let's examine the given statements:
The focus is located at ( 0 , − 3 ) . - This is true.
The parabola opens to the left. - This is false, as it opens downwards.
The p value can be determined by computing 4(3). - This is false; p is 3 itself, not 4(3).
The parabola can be represented by the equation x 2 = − 12 y . - This is true.
The parabola can be represented by the equation y 2 = 12 x . - This is false.
Examples
Parabolas are commonly seen in the real world, such as in the design of satellite dishes and suspension bridges. The reflective property of parabolas is used in satellite dishes to focus incoming signals onto a single point. Similarly, understanding the properties of parabolas helps engineers design suspension bridges that can distribute weight evenly. In sports, the trajectory of a ball (like a basketball or soccer ball) when thrown or kicked often approximates a parabolic path, especially when air resistance is negligible. Knowing the vertex and focus of this parabolic path can help athletes optimize their throws or kicks for maximum distance or accuracy.
The focus of the parabola is at (0, -3) and its equation is x 2 = − 12 y . Therefore, the correct statements are: A. The focus is located at (0,-3) and D. The parabola can be represented by the equation x 2 = − 12 y .
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