Lauren describes a parabola where the focus has a positive, nonzero [tex]$x$[/tex] coordinate. Which parabola(s) could Lauren be describing? Check all that apply.
[tex]$x^2=4 y$[/tex]
[tex]$x^2=-6 y$[/tex]
[tex]$y^2=x$[/tex]
[tex]$y^2=10 x$[/tex]
[tex]$y^2=-3 x$[/tex]
[tex]$y^2=5 x$[/tex]
Answer (1)
Analyze the standard forms of parabolas and their foci.
Determine the p value for each given parabola.
Identify the parabolas with a focus that has a positive, nonzero x coordinate.
The parabolas are: y 2 = x , y 2 = 10 x , y 2 = 5 x .
Explanation
Problem Analysis Let's analyze the problem. We are given several parabolas and need to determine which ones have a focus with a positive, nonzero x coordinate. Recall that the standard form of a parabola that opens to the right or left is y 2 = 4 p x , where the focus is at ( p , 0 ) . The standard form of a parabola that opens upwards or downwards is x 2 = 4 p y , where the focus is at ( 0 , p ) . We need to find the parabolas where 0"> p > 0 and the focus is on the x -axis.
Examine Each Parabola Now let's examine each parabola:
x 2 = 4 y : This is of the form x 2 = 4 p y , so 4 p = 4 , which means p = 1 . The focus is at ( 0 , 1 ) . The x -coordinate is 0, so this parabola does not satisfy the condition.
x 2 = − 6 y : This is of the form x 2 = 4 p y , so 4 p = − 6 , which means p = − 2 3 . The focus is at ( 0 , − 2 3 ) . The x -coordinate is 0, so this parabola does not satisfy the condition.
y 2 = x : This is of the form y 2 = 4 p x , so 4 p = 1 , which means p = 4 1 . The focus is at ( 4 1 , 0 ) . The x -coordinate is positive and nonzero, so this parabola satisfies the condition.
y 2 = 10 x : This is of the form y 2 = 4 p x , so 4 p = 10 , which means p = 2 5 . The focus is at ( 2 5 , 0 ) . The x -coordinate is positive and nonzero, so this parabola satisfies the condition.
y 2 = − 3 x : This is of the form y 2 = 4 p x , so 4 p = − 3 , which means p = − 4 3 . The focus is at ( − 4 3 , 0 ) . The x -coordinate is negative, so this parabola does not satisfy the condition.
y 2 = 5 x : This is of the form y 2 = 4 p x , so 4 p = 5 , which means p = 4 5 . The focus is at ( 4 5 , 0 ) . The x -coordinate is positive and nonzero, so this parabola satisfies the condition.
Final Answer The parabolas that have a focus with a positive, nonzero x coordinate are y 2 = x , y 2 = 10 x , and y 2 = 5 x .
Examples
Understanding parabolas and their foci is crucial in various fields, such as satellite dish design. Satellite dishes are shaped like paraboloids, and the receiver is placed at the focus of the parabola. This design ensures that incoming signals are reflected and concentrated at the receiver, maximizing signal strength. Knowing the properties of parabolas allows engineers to optimize the placement and shape of satellite dishes for efficient signal reception.
