Question

3
use the distance formula to write an equation of the parabola.
an equation of the parabola is y =

Explanation:

Step1: Recall the definition of a parabola

A parabola is the set of all points \(P(x,y)\) such that the distance from \(P\) to the focus \(F(0,1)\) is equal to the distance from \(P\) to the directrix \(y = - 1\).
The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
The distance from \(P(x,y)\) to \(F(0,1)\) is \(d_{PF}=\sqrt{(x - 0)^2+(y - 1)^2}\).
The distance from \(P(x,y)\) to \(D(x,-1)\) (a point on the directrix \(y=-1\)) is \(d_{PD}=\sqrt{(x - x)^2+(y+1)^2}=\vert y + 1\vert\).

Step2: Set up the equation

Since \(d_{PF}=d_{PD}\), we have \(\sqrt{x^{2}+(y - 1)^{2}}=\vert y + 1\vert\).
Square both sides of the equation: \(x^{2}+(y - 1)^{2}=(y + 1)^{2}\).

Step3: Expand and simplify

Expand \((y - 1)^{2}=y^{2}-2y + 1\) and \((y + 1)^{2}=y^{2}+2y + 1\).
So \(x^{2}+y^{2}-2y + 1=y^{2}+2y + 1\).
Subtract \(y^{2}+1\) from both sides: \(x^{2}-2y=2y\).
Then \(x^{2}=4y\), and \(y=\frac{1}{4}x^{2}\).

Answer:

\(y=\frac{1}{4}x^{2}\)