Find the standard form of the equation of the parabola with the given characteristics and vertex at the origin. Directrix: x = -4.

Solution:

Given that the vertex of the parabola is centered at the origin and given that directrix is at x = -4, one can state that the focus is at (4,0).

The standard form of the equation can therefore be derived by assuming a point P(x,y) on the parabola which is equidistant from the directrix and the focus.

Therefore we can write:

\(\sqrt{(x - 4)^{2} + (y-0)^{2}} = \sqrt{(x-(-4))^{2} + (y-y)^{2}}\)

\(\sqrt{(x - 4)^{2} + (y)^{2}} = \sqrt{(x+4)^{2} + (0)^{2}}\)

Squaring both sides we get

\((x - 4)^{2} + (y)^{2} = (x+4)^{2}\)

x² - 8x + 16 + y²  =  x² + 8x + 16

y²  = 16x

x = y² /16

Find the standard form of the equation of the parabola with the given characteristics and vertex at the origin. Directrix: x = -4.

Summary:

The standard form of the equation of the parabola with the given characteristics and vertex at the origin & Directrix: x = -4 is x = y² /16.