Question

use the distance formula to write an equation of the parabola with focus (0, - 2) and directrix y = 2. an equation of the parabola is y = square.

Explanation:

Step1: Recall the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For a parabola, the distance from any point \((x,y)\) on the parabola to the focus \((0,-2)\) is equal to the distance from the point \((x,y)\) to the directrix \(y = 2\).

Step2: Set up the distance equations

The distance from \((x,y)\) to \((0,-2)\) is \(d_1=\sqrt{(x - 0)^2+(y+ 2)^2}\), and the distance from \((x,y)\) to the line \(y = 2\) (using the formula \(d=\vert y - 2\vert\) for the distance from a point \((x,y)\) to the horizontal line \(y = k\)) is \(d_2=\vert y - 2\vert\).

Step3: Square both sides of the equation

Since \(d_1=d_2\), we have \((x - 0)^2+(y + 2)^2=(y - 2)^2\).

Step4: Expand the equations

Expand \((y + 2)^2=y^{2}+4y + 4\) and \((y - 2)^2=y^{2}-4y + 4\). Then \(x^{2}+y^{2}+4y + 4=y^{2}-4y + 4\).

Step5: Simplify the equation

Subtract \(y^{2}+4\) from both sides. We get \(x^{2}+4y=-4y\). Then \(x^{2}=-8y\), and solving for \(y\) gives \(y=-\frac{1}{8}x^{2}\).

Answer:

\(y =-\frac{1}{8}x^{2}\)