The derivation for the equation of a parabola with a vertex at the origin is started below.
[tex]\sqrt{(x-0)^2+(y-p)^2}=\sqrt{(x-x)^2+(y-(-p))^2}[/tex]
1. [tex](x)^2+(y-p)^2=(0)^2+(y+p)^2[/tex]
2. [tex]x^2+y^2-2 p y+p^2=y^2+2 p y+p^2[/tex]
If the equation is further simplified, which equation for a parabola does it become?
A. [tex]x^2=4 p y[/tex]
B. [tex]x^2=2 y^2+2 p^2[/tex]
C. [tex]y^2=4 p x[/tex]
D. [tex]y^2=4 p y[/tex]
Answer (1)
Start with the given equation x 2 + y 2 − 2 p y + p 2 = y 2 + 2 p y + p 2 .
Simplify by subtracting y 2 and p 2 from both sides, resulting in x 2 − 2 p y = 2 p y .
Add 2 p y to both sides to isolate x 2 , which gives x 2 = 4 p y .
The simplified equation for the parabola is x 2 = 4 p y .
Explanation
Analyze the given equation. We are given the equation x 2 + y 2 − 2 p y + p 2 = y 2 + 2 p y + p 2 and we want to simplify it to find the equation of a parabola.
Subtract y 2 from both sides. Subtract y 2 from both sides of the equation: x 2 + y 2 − 2 p y + p 2 − y 2 = y 2 + 2 p y + p 2 − y 2 , which simplifies to x 2 − 2 p y + p 2 = 2 p y + p 2 .
Subtract p 2 from both sides. Subtract p 2 from both sides of the equation: x 2 − 2 p y + p 2 − p 2 = 2 p y + p 2 − p 2 , which simplifies to x 2 − 2 p y = 2 p y .
Add 2 p y to both sides. Add 2 p y to both sides of the equation: x 2 − 2 p y + 2 p y = 2 p y + 2 p y , which simplifies to x 2 = 4 p y .
State the final equation. The simplified equation is x 2 = 4 p y , which matches the first option.
Examples
Parabolas are commonly found in the real world, such as the trajectory of a ball thrown in the air or the shape of a satellite dish. Understanding the equation of a parabola allows us to model and analyze these phenomena. For example, if we know the value of p , we can determine the width and depth of a satellite dish needed to focus signals effectively. Similarly, we can predict the landing point of a projectile if we know its initial velocity and angle, using the parabolic trajectory equation.
