Question

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 ).
( sqrt{(x - p)^2 + (y - 0)^2} = sqrt{(x - (-p))^2 + (y - y)^2} )
( y^2 = )
options: -4px, 0, 2px, 4px

Explanation:

Step1: Square both sides

To eliminate the square roots, we square both sides of the equation \(\sqrt{(x - p)^2 + (y - 0)^2}=\sqrt{(x - (-p))^2+(y - y)^2}\).
Squaring the left side gives \((x - p)^2 + y^2\), and squaring the right side gives \((x + p)^2+(y - y)^2=(x + p)^2+0=(x + p)^2\). So we have \((x - p)^2 + y^2=(x + p)^2\).

Step2: Expand the squares

Expand \((x - p)^2\) and \((x + p)^2\) using the formula \((a\pm b)^2=a^2\pm2ab + b^2\).
\((x - p)^2=x^2-2px + p^2\) and \((x + p)^2=x^2+2px + p^2\). Substituting these into the equation, we get \(x^2-2px + p^2 + y^2=x^2+2px + p^2\).

Step3: Simplify the equation

Subtract \(x^2\) and \(p^2\) from both sides of the equation.
\(x^2-2px + p^2 + y^2-x^2 - p^2=x^2+2px + p^2-x^2 - p^2\), which simplifies to \(-2px + y^2=2px\).

Step4: Solve for \(y^2\)

Add \(2px\) to both sides of the equation \(-2px + y^2=2px\).
\(-2px + y^2+2px=2px + 2px\), so \(y^2 = 4px\).

Answer:

\(4px\)