The parabola opens to the right. The focus is given as $F(p, 0)$ and directrix $x=-p$. The distance between the focus and point $P$ is equal to the distance between the directrix and point $P$. Continue to simplify the equation to solve for $y^2$.
$\begin{array}{c}
F P=Q P \\
\sqrt{(x-p)^2+(y-0)^2}=\sqrt{(x-(-p))^2+(y-y)^2}
\end{array}$
$y^2 = $
$\square$
Answer (1)
Start with the equation ( x − p ) 2 + ( y − 0 ) 2 = ( x − ( − p ) ) 2 + ( y − y ) 2 .
Square both sides to get ( x − p ) 2 + y 2 = ( x + p ) 2 .
Expand and simplify the equation: x 2 − 2 p x + p 2 + y 2 = x 2 + 2 p x + p 2 .
Isolate y 2 to find the solution: y 2 = 4 p x .
Explanation
Problem Analysis The problem states that a parabola opens to the right with focus F ( p , 0 ) and directrix x = − p . We are given the equation representing the equality of distances between a point P ( x , y ) on the parabola and the focus F , and between the point P and the directrix. Our goal is to simplify this equation and solve for y 2 .
Squaring both sides We start with the given equation: ( x − p ) 2 + ( y − 0 ) 2 = ( x − ( − p ) ) 2 + ( y − y ) 2 To eliminate the square roots, we square both sides of the equation: ( ( x − p ) 2 + ( y − 0 ) 2 ) 2 = ( ( x + p ) 2 + ( y − y ) 2 ) 2 This simplifies to: ( x − p ) 2 + y 2 = ( x + p ) 2
Expanding the terms Next, we expand the squared terms: x 2 − 2 p x + p 2 + y 2 = x 2 + 2 p x + p 2
Isolating y 2 Now, we want to isolate y 2 . We subtract x 2 , p 2 and add 2 p x from both sides of the equation: y 2 = x 2 + 2 p x + p 2 − x 2 + 2 p x − p 2
Simplifying the equation Finally, we simplify the equation to find y 2 :
y 2 = 4 p x
Final Answer Thus, we have found that y 2 = 4 p x .
Examples
Parabolas are essential in designing satellite dishes and reflecting telescopes. The reflective property of a parabola ensures that signals or light rays coming from a distant source are focused at a single point (the focus). Knowing the equation y 2 = 4 p x allows engineers to precisely determine the shape of the dish needed to focus signals efficiently, maximizing signal strength and clarity.
