Question

use the distance formula to write an equation of the parabola with vertex (0,0) and focus (0, - 10). an equation of the parabola is y = □.

Explanation:

Step1: Recall the standard form of a parabola

For a parabola with vertex \((h,k)\) and focus \((h,k + p)\), the standard form is \((x - h)^2=4p(y - k)\). Here, \(h = 0,k = 0\) (vertex \((0,0)\)) and the focus is \((0,- 10)\), so \(k + p=-10\). Since \(k = 0\), then \(p=-10\).

Step2: Substitute values into the standard form

Substitute \(h = 0,k = 0,p=-10\) into \((x - h)^2=4p(y - k)\). We get \(x^{2}=4\times(-10)(y - 0)\).

Answer:

\(x^{2}=-40y\)