Answer:
To find the equation of a parabola given its focus and directrix, we can use the definition of a parabola.
The standard form of the equation of a parabola with its focus at (h, k + p) and directrix y = k - p is:
(x - h)^2 = 4p(y - k)
In this case, the focus is (2, 4) and the directrix is x = 10. Since the directrix is a vertical line, we have a horizontal parabola.
Comparing the given values to the standard form, we have:
h = 2
k + p = 4
k - p = 10
From the equations k + p = 4 and k - p = 10, we can solve for k and p by adding the two equations together:
(k + p) + (k - p) = 4 + 10
2k = 14
k = 7
Substituting the value of k into either equation, we can solve for p:
7 + p = 4
p = 4 - 7
p = -3
Now we have the values of h, k, and p. Substituting them into the standard form equation, we get:
(x - 2)^2 = 4(-3)(y - 7)
Simplifying further:
(x - 2)^2 = -12(y - 7)
Thus, the equation of the parabola with focus (2, 4) and directrix x = 10 is (x - 2)^2 = -12(y - 7).