A general formula for a parabola is [tex]$x^2=4 p y$[/tex]. What is the value of [tex]$p$[/tex] in the equation [tex]$x^2=12 y$[/tex] ?
A. [tex]$p=3$[/tex]
B. [tex]$p=4$[/tex]
C. [tex]$p=6$[/tex]
D. [tex]$p=12$[/tex]
Answer (1)
Compare the given equation x 2 = 12 y with the general form x 2 = 4 p y .
Equate the coefficients of y : 4 p = 12 .
Solve for p by dividing both sides by 4: p = 4 12 = 3 .
The value of p is 3 .
Explanation
Understanding the Problem We are given the general form of a parabola as x 2 = 4 p y and a specific instance x 2 = 12 y . Our goal is to find the value of p in the given equation.
Setting up the Equation To find the value of p , we need to compare the given equation x 2 = 12 y with the general form x 2 = 4 p y . By comparing the two equations, we can see that the coefficient of y in the general form is 4 p , and in the given equation, it is 12 . Therefore, we can set up the equation 4 p = 12 .
Solving for p Now, we solve the equation 4 p = 12 for p . To do this, we divide both sides of the equation by 4 : 4 4 p = 4 12 p = 3 Thus, the value of p is 3 .
Examples
Understanding parabolas is crucial in various fields, such as physics and engineering. For example, the trajectory of a projectile (like a ball thrown in the air) follows a parabolic path. If we know the equation of this path, we can determine key parameters like the maximum height reached or the distance traveled. Similarly, parabolic reflectors are used in satellite dishes and car headlights to focus signals or light, and the value of 'p' helps determine the shape and focal point of these reflectors.
