Find the equation of each of the following parabolas

Directrix x = 0, focus at (6, 0)

the distance of any point on the parabola from its focus and its directrix is same.

Given that, directrix, x = 0 and focus = (6, 0)

If a parabola has a vertical axis, the standard form of the equation of the parabola is (x - h)² = 4p(y - k), where p≠ 0.

The vertex of this parabola is at (h, k).

The focus is at (h, k + p) & the directrix is the line y = k - p.

As the focus lies on x – axis,

Equation is y² = 4ax or y² = -4ax

So, for any point P(x, y) on the parabola

Distance of point from directrix = Distance of point from focus

x² = (x – 6)² + y²

x² = x² - 12x + 36 + y²

y² - 12x + 36 = 0

Hence the required equation is y² - 12x + 36 = 0.