Describe the key features of a parabola with the equation [tex]x^2=40 y[/tex].
The value of [tex]p[/tex] is $\square$.
The parabola opens $\square$.
The coordinates of the focus are $\square$.
The equation for the directrix is $\square$.
Answer (2)
The value of p for the parabola x 2 = 40 y is 10, and it opens upwards. The focus is located at the coordinates ( 0 , 10 ) , while the directrix is represented by the equation y = − 10 .
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Determine the value of p by comparing the given equation x 2 = 40 y to the standard form x 2 = 4 p y , which gives 4 p = 40 , so p = 10 .
Since 0"> p > 0 , the parabola opens upwards.
Identify the focus at ( 0 , p ) , which is ( 0 , 10 ) .
Determine the directrix equation as y = − p , which is y = − 10 .
p = 10 , opens upwards, focus ( 0 , 10 ) , directrix y = − 10
Explanation
Problem Analysis We are given the equation of a parabola as x 2 = 40 y . Our goal is to determine the value of p , the direction the parabola opens, the coordinates of the focus, and the equation of the directrix.
Relate to Standard Form The standard form of a parabola that opens upwards or downwards is x 2 = 4 p y , where p is the distance from the vertex to the focus and from the vertex to the directrix. Comparing the given equation x 2 = 40 y with the standard form x 2 = 4 p y , we can equate the coefficients of y to find the value of p .
Solve for p We have 4 p = 40 . Dividing both sides by 4, we get p = 4 40 = 10 .
Determine Direction Since p = 10 is positive, the parabola opens upwards.
Find Focus Coordinates For a parabola in the form x 2 = 4 p y that opens upwards, the focus is at the point ( 0 , p ) . Since p = 10 , the coordinates of the focus are ( 0 , 10 ) .
Find Directrix Equation The equation of the directrix for a parabola in the form x 2 = 4 p y is y = − p . Since p = 10 , the equation of the directrix is y = − 10 .
Final Answer In summary, for the parabola x 2 = 40 y , we have:
The value of p is 10.
The parabola opens upwards.
The coordinates of the focus are ( 0 , 10 ) .
The equation for the directrix is y = − 10 .
Examples
Parabolas are commonly seen in the design of satellite dishes and reflective telescopes. The parabolic shape helps to focus incoming signals or light to a single point (the focus), which is where the receiver or detector is placed. Understanding the properties of a parabola, such as the location of its focus and directrix, is crucial in optimizing the design and performance of these devices. For example, if you're building a solar oven with a parabolic reflector, knowing the focus helps you position your cooking pot for maximum heat.
