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
$(y-k)^2=4 p(x-h)$
$(x-h)^2=4 p(y-k)$
The vertex is given at $(2,-1)$, which tells us that $h=$ $\square$ , and $k=$ $\square$
Answer (1)
Identify the vertex ( h , k ) as ( 2 , − 1 ) , so h = 2 and k = − 1 .
Determine that the parabola opens horizontally since the directrix is a vertical line x = 5 , giving the form ( y − k ) 2 = 4 p ( x − h ) .
Calculate p as the directed distance from the vertex to the directrix: p = − 3 (negative because the parabola opens to the left).
Substitute h , k , and p into the parabola equation: ( y + 1 ) 2 = − 12 ( x − 2 ) .
Explanation
Identify h and k from the vertex The vertex of the parabola is given as ( 2 , − 1 ) . This directly tells us the values of h and k , which are the x and y coordinates of the vertex, respectively. Therefore, h = 2 and k = − 1 .
Determine the form of the parabola The directrix is given as x = 5 . Since the directrix is a vertical line, 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 and p is the directed distance from the vertex to the focus (or from the vertex to the directrix).
Calculate p The distance between the vertex ( 2 , − 1 ) and the directrix x = 5 is ∣5 − 2∣ = 3 . Since the directrix is to the right of the vertex, the parabola opens to the left. This means that p is negative, so p = − 3 .
Write the equation of the parabola Now we 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 )
State the final answer The equation of the parabola with vertex ( 2 , − 1 ) and directrix x = 5 is ( y + 1 ) 2 = − 12 ( x − 2 ) .
Examples
Parabolas are commonly used in the design of satellite dishes and reflective telescopes. The parabolic shape helps to focus incoming signals or light to a single point, which is crucial for effective communication and observation. Understanding how to determine the equation of a parabola given its vertex and directrix allows engineers to design these devices with precision, ensuring optimal performance.
