A general formula for a parabola is [tex]$y^2=4 p x$[/tex]. What is the value of [tex]$p$[/tex] in the equation [tex]$y^2=-4 x$[/tex]?
A. [tex]$P=-4$[/tex]
B. [tex]$P=-1$[/tex]
C. [tex]$P=1$[/tex]
D. [tex]$P=4$[/tex]
Answer (1)
Compare the given equation y 2 = − 4 x with the general form y 2 = 4 p x .
Equate the coefficients of x : 4 p = − 4 .
Solve for p : p = 4 − 4 = − 1 .
The value of p is − 1 .
Explanation
Understanding the Problem We are given the general formula for a parabola as y 2 = 4 p x . We are also given the equation y 2 = − 4 x . 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 x in the general formula is 4 p , and the coefficient of x in the given equation is − 4 . Therefore, we can set up the equation 4 p = − 4 .
Solving for p Now, we solve the equation 4 p = − 4 for p . To do this, we divide both sides of the equation by 4: 4 4 p = 4 − 4 p = − 1
Final Answer Therefore, the value of p in the equation y 2 = − 4 x is − 1 .
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. By knowing the equation of the parabola, we can determine the range and maximum height of the projectile. Similarly, parabolic reflectors are used in satellite dishes and car headlights to focus signals or light into a specific point. The value of p in the equation helps determine the shape and focus of the parabola, which is essential for designing these devices.
