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 .
The value of p is 3 .
Explanation
Understanding the Problem We are given the general formula for a parabola as x 2 = 4 p y and a specific equation 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 with the general formula. We can see that the coefficient of y in the general formula 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 for p by dividing both sides of the equation 4 p = 12 by 4 . This gives us: p = 4 12 = 3 So, the value of p is 3 .
Final Answer Therefore, the value of p in the equation x 2 = 12 y is 3 .
Examples
Understanding parabolas is crucial in various fields like physics and engineering. For instance, the trajectory of a projectile, like a ball thrown in the air, follows a parabolic path. The value of p in the equation determines the shape and focus of the parabola, which helps in predicting the range and height of the projectile. Similarly, satellite dishes and reflecting telescopes use parabolic reflectors to focus signals or light to a single point, and the value of p is essential in designing these devices.
