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 (2)
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 .
The value of p is − 1 .
Explanation
Problem Analysis We are given the general formula for a parabola as y 2 = 4 p x and a specific equation y 2 = − 4 x . Our goal is to find the value of p in the given equation.
Equating Coefficients To find the value of p , we need to compare the given equation with the general formula. We can equate the coefficients of x in both equations.
Setting up the Equation Comparing y 2 = 4 p x with y 2 = − 4 x , we can write:
4 p = − 4
Solving for p Now, we solve for p by dividing both sides of the equation by 4:
p = 4 − 4 = − 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, such as 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 of the parabola helps determine the shape and orientation of this path, which is essential for predicting the projectile's range and maximum height. Similarly, parabolic reflectors are used in satellite dishes and solar cookers to focus incoming signals or sunlight onto a single point, and the value of 'p' determines the focal length of these reflectors.
The value of p in the equation y 2 = − 4 x is − 1 . This is determined by comparing the equation to the general form of a parabola, y 2 = 4 p x , and equating the coefficients. Hence, the correct answer is option B: p = − 1 .
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