The shape of a satellite dish can be described as parabolic. Satellite dishes are this shape because radio waves are reflected from the surface of the dish and received into the focus. If the graph of the satellite dish is given by the equation [tex]$x^2=8 y$[/tex], what are the coordinates of the focus?
([tex]$\square$[/tex], [tex]$\square$[/tex])
Answer (1)
The equation of the parabola is x 2 = 8 y .
Comparing with the standard form x 2 = 4 a y , we find 4 a = 8 .
Solving for a , we get a = 2 .
The focus of the parabola is at ( 0 , a ) , so the coordinates of the focus are ( 0 , 2 ) .
Explanation
Analyze the problem The equation of the satellite dish is given by x 2 = 8 y . We need to find the coordinates of the focus of this parabola.
Recall the standard form of a parabola The standard form of a parabola with its vertex at the origin and opening upwards is given by x 2 = 4 a y , where a is the distance from the vertex to the focus.
Compare the given equation with the standard form Comparing the given equation x 2 = 8 y with the standard form x 2 = 4 a y , we can write:
4 a = 8
Solve for a Solving for a , we divide both sides of the equation 4 a = 8 by 4:
a = 4 8 = 2
Determine the coordinates of the focus Since the vertex of the parabola is at the origin (0, 0) and the parabola opens upwards, the focus is at the point (0, a). Substituting a = 2 , the coordinates of the focus are (0, 2).
State the final answer Therefore, the coordinates of the focus of the satellite dish are (0, 2).
Examples
Understanding the focus of a parabola is crucial in designing satellite dishes and solar collectors. The parabolic shape concentrates incoming signals or sunlight onto the focus, maximizing efficiency. For instance, if you're building a solar oven, knowing the focus point helps you position the cooking pot for optimal heating.
