Write the equation for a parabola that has a vertex $(2,-1)$ and a directrix of $x=5$. The standard form for this parabola is
$(y-k)^2=4 p(x-h)$
$(x-h)^2=4 p(y-k)$
Answer (1)
Determine that the parabola opens horizontally since the directrix is a vertical line.
Identify the vertex as ( h , k ) = ( 2 , − 1 ) .
Calculate p = − 3 as the directed distance from the vertex to the focus.
Substitute h , k , and p into the equation ( y − k ) 2 = 4 p ( x − h ) to get ( y + 1 ) 2 = − 12 ( x − 2 ) .
Explanation
Understanding the Problem We are given the vertex of the parabola as ( 2 , − 1 ) and the directrix as x = 5 . We need to find the equation of the parabola.
Determining the Parabola's Orientation Since the directrix is a vertical line x = 5 , the parabola opens either to the left or to the right. The general form of such a parabola is ( y − k ) 2 = 4 p ( x − h ) , where ( h , k ) is the vertex of the parabola and p is the directed distance from the vertex to the focus.
Finding the Value of p The vertex is given as ( 2 , − 1 ) , so h = 2 and k = − 1 . The directrix is x = 5 . Since the directrix is to the right of the vertex, the parabola opens to the left. The distance between the vertex and the directrix is ∣5 − 2∣ = 3 . Therefore, p = − 3 because the parabola opens to the left.
Substituting the Values Now, substitute the values of h , k , and p into the equation ( y − k ) 2 = 4 p ( x − h ) : ( y − ( − 1 ) ) 2 = 4 ( − 3 ) ( x − 2 ) ( y + 1 ) 2 = − 12 ( x − 2 )
Final Equation Thus, the equation of the parabola is ( y + 1 ) 2 = − 12 ( x − 2 ) .
Examples
Parabolas are commonly seen in real-world applications such as satellite dishes and suspension bridges. The reflective property of parabolas is used in satellite dishes to focus incoming signals onto a single point. Similarly, the parabolic shape of suspension bridge cables helps distribute weight evenly. Understanding the equation of a parabola allows engineers to design these structures efficiently.
