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 (1)
The vertex is at the origin and the directrix is y = 3 , so the parabola opens downwards.
The focus is at ( 0 , − 3 ) , and p = − 3 .
The equation of the parabola is x 2 = 4 p y = − 12 y .
The true statements are: the focus is at ( 0 , − 3 ) and the equation is x 2 = − 12 y .
The focus is located at ( 0 , − 3 ) and x 2 = − 12 y
Explanation
Analyze the Parabola The vertex of the parabola is at the origin ( 0 , 0 ) , and the directrix is given by 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 and p-value The focus of a parabola is located at a distance ∣ p ∣ from the vertex, in the opposite direction of the directrix. Since the directrix is y = 3 , the focus is at ( 0 , − 3 ) . Thus, p = − 3 .
Find the Equation of the Parabola The general equation for a parabola that opens upwards or 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
x 2 = − 12 y
Evaluate the Statements Now, let's evaluate the given statements:
The focus is located at ( 0 , − 3 ) . This statement is true.
The parabola opens to the left. This statement is false, as the parabola opens downwards.
The p value can be determined by computing 4(3). This statement is false. The value of p is − 3 , and 4∣ p ∣ is used in the equation, not to determine p .
The parabola can be represented by the equation x 2 = − 12 y . This statement is true.
The parabola can be represented by the equation y 2 = 12 x . This statement is false, as this equation represents a parabola opening to the right.
Final Answer 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 .
Examples
Parabolas are commonly used in the design of satellite dishes and reflecting telescopes. The shape of a parabola allows incoming signals (e.g., radio waves or light) to be focused at a single point, the focus, where a receiver is placed. Knowing the position of the directrix helps in determining the optimal placement and shape of the parabolic reflector to efficiently collect and concentrate the signals.
