Question

use the distance formula to write an equation of the parabola with vertex (0,0) and directrix y = -6. an equation of the parabola is y = □.

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 with vertex \((h,k)=(0,0)\), the definition of a parabola is the set of points \((x,y)\) such that the distance from the point \((x,y)\) to the focus \((0,p)\) is equal to the distance from the point \((x,y)\) to the directrix \(y=-p\).

Step2: Find the value of \(p\)

The directrix is \(y = - 9\). For a parabola \(y=ax^{2}\) (vertex at \((0,0)\)), the directrix is \(y=-p\). Here \(p = 9\) (since the distance from the vertex \((0,0)\) to the directrix \(y=-9\) is \(|0-(-9)|=9\)).

Step3: Use the standard form of the parabola equation

The standard form of a parabola with vertex \((h,k)=(0,0)\) and focus \((0,p)\) is \(x^{2}=4py\). Substituting \(p = 9\), we get \(x^{2}=4\times9y\), which simplifies to \(y=\frac{1}{36}x^{2}\).

Answer:

\(y=\frac{1}{36}x^{2}\)